Mister Exam
Other calculators
-
How to use it?
- Equation:
-
Equation x^2+6x-7=0
-
Equation (x+3)^2=4
-
Equation 2*x^2-18=0
-
Equation x*x-x+7=0
- Express {x} in terms of y where:
- -15*x+3*y=8
- -1*x+11*y=-9
- 1*x+18*y=-5
- 15*x+9*y=9
-
Identical expressions
- eight *x^ two + eight *y^ two - sixteen *a*(x-y)+ fifteen *a^ two - forty-eight *y- fifty *a+ seventy-one = zero
- 8 multiply by x squared plus 8 multiply by y squared minus 16 multiply by a multiply by (x minus y) plus 15 multiply by a squared minus 48 multiply by y minus 50 multiply by a plus 71 equally 0
- eight multiply by x to the power of two plus eight multiply by y to the power of two minus sixteen multiply by a multiply by (x minus y) plus fifteen multiply by a to the power of two minus forty minus eight multiply by y minus fifty multiply by a plus seventy minus one equally zero
- 8*x2+8*y2-16*a*(x-y)+15*a2-48*y-50*a+71=0
- 8*x2+8*y2-16*a*x-y+15*a2-48*y-50*a+71=0
- 8*x²+8*y²-16*a*(x-y)+15*a²-48*y-50*a+71=0
- 8*x to the power of 2+8*y to the power of 2-16*a*(x-y)+15*a to the power of 2-48*y-50*a+71=0
- 8x^2+8y^2-16a(x-y)+15a^2-48y-50a+71=0
- 8x2+8y2-16a(x-y)+15a2-48y-50a+71=0
- 8x2+8y2-16ax-y+15a2-48y-50a+71=0
- 8x^2+8y^2-16ax-y+15a^2-48y-50a+71=0
- 8*x^2+8*y^2-16*a*(x-y)+15*a^2-48*y-50*a+71=O
-
Similar expressions
- 8*x^2+8*y^2-16*a*(x-y)+15*a^2+48*y-50*a+71=0
- 8*x^2+8*y^2-16*a*(x-y)-15*a^2-48*y-50*a+71=0
- 8*x^2+8*y^2-16*a*(x+y)+15*a^2-48*y-50*a+71=0
- 8*x^2+8*y^2-16*a*(x-y)+15*a^2-48*y-50*a-71=0
- 8*x^2-8*y^2-16*a*(x-y)+15*a^2-48*y-50*a+71=0
- 8*x^2+8*y^2+16*a*(x-y)+15*a^2-48*y-50*a+71=0
- 8*x^2+8*y^2-16*a*(x-y)+15*a^2-48*y+50*a+71=0
8*x^2+8*y^2-16*a*(x-y)+15*a^2-48*y-50*a+71=0 equation
The teacher will be very surprised to see your correct solution 😉
The solution
You have entered
[src]
2 2 2 8*x + 8*y - 16*a*(x - y) + 15*a - 48*y - 50*a + 71 = 0
$$\left(- 50 a + \left(- 48 y + \left(15 a^{2} + \left(- 16 a \left(x - y\right) + \left(8 x^{2} + 8 y^{2}\right)\right)\right)\right)\right) + 71 = 0$$
Detail solution
Expand the expression in the equation
$$\left(- 50 a + \left(- 48 y + \left(15 a^{2} + \left(- 16 a \left(x - y\right) + \left(8 x^{2} + 8 y^{2}\right)\right)\right)\right)\right) + 71 = 0$$
We get the quadratic equation
$$15 a^{2} - 16 a x + 16 a y - 50 a + 8 x^{2} + 8 y^{2} - 48 y + 71 = 0$$
This equation is of the form
A quadratic equation can be solved
using the discriminant.
The roots of the quadratic equation:
$$x_{1} = \frac{\sqrt{D} - b}{2 a}$$
$$x_{2} = \frac{- \sqrt{D} - b}{2 a}$$
where D = b^2 - 4*a*c - it is the discriminant.
Because
$$a = 8$$
$$b = - 16 a$$
$$c = 15 a^{2} + 16 a y - 50 a + 8 y^{2} - 48 y + 71$$
, then
The equation has two roots.
or
$$x_{1} = a + \frac{\sqrt{- 224 a^{2} - 512 a y + 1600 a - 256 y^{2} + 1536 y - 2272}}{16}$$
$$x_{2} = a - \frac{\sqrt{- 224 a^{2} - 512 a y + 1600 a - 256 y^{2} + 1536 y - 2272}}{16}$$
$$\left(- 50 a + \left(- 48 y + \left(15 a^{2} + \left(- 16 a \left(x - y\right) + \left(8 x^{2} + 8 y^{2}\right)\right)\right)\right)\right) + 71 = 0$$
We get the quadratic equation
$$15 a^{2} - 16 a x + 16 a y - 50 a + 8 x^{2} + 8 y^{2} - 48 y + 71 = 0$$
This equation is of the form
a*x^2 + b*x + c = 0
A quadratic equation can be solved
using the discriminant.
The roots of the quadratic equation:
$$x_{1} = \frac{\sqrt{D} - b}{2 a}$$
$$x_{2} = \frac{- \sqrt{D} - b}{2 a}$$
where D = b^2 - 4*a*c - it is the discriminant.
Because
$$a = 8$$
$$b = - 16 a$$
$$c = 15 a^{2} + 16 a y - 50 a + 8 y^{2} - 48 y + 71$$
, then
D = b^2 - 4 * a * c =
(-16*a)^2 - 4 * (8) * (71 - 50*a - 48*y + 8*y^2 + 15*a^2 + 16*a*y) = -2272 - 256*y^2 - 224*a^2 + 1536*y + 1600*a - 512*a*y
The equation has two roots.
x1 = (-b + sqrt(D)) / (2*a)
x2 = (-b - sqrt(D)) / (2*a)
or
$$x_{1} = a + \frac{\sqrt{- 224 a^{2} - 512 a y + 1600 a - 256 y^{2} + 1536 y - 2272}}{16}$$
$$x_{2} = a - \frac{\sqrt{- 224 a^{2} - 512 a y + 1600 a - 256 y^{2} + 1536 y - 2272}}{16}$$
The graph
Rapid solution
[src]
/ _______________________________________________________________________________________________________________________________________________________________________ \ _______________________________________________________________________________________________________________________________________________________________________
| / 2 / / 2 2 2 2 \\ | / 2 / / 2 2 2 2 \\
| 4 / 2 / 2 2 2 2 \ |atan2\-32*im(a*y) + 96*im(y) + 100*im(a) - 32*im(y)*re(y) - 28*im(a)*re(a), -142 - 32*re(a*y) - 16*re (y) - 14*re (a) + 14*im (a) + 16*im (y) + 96*re(y) + 100*re(a)/| | 4 / 2 / 2 2 2 2 \ |atan2\-32*im(a*y) + 96*im(y) + 100*im(a) - 32*im(y)*re(y) - 28*im(a)*re(a), -142 - 32*re(a*y) - 16*re (y) - 14*re (a) + 14*im (a) + 16*im (y) + 96*re(y) + 100*re(a)/|
| \/ (-32*im(a*y) + 96*im(y) + 100*im(a) - 32*im(y)*re(y) - 28*im(a)*re(a)) + \-142 - 32*re(a*y) - 16*re (y) - 14*re (a) + 14*im (a) + 16*im (y) + 96*re(y) + 100*re(a)/ *sin|---------------------------------------------------------------------------------------------------------------------------------------------------------------------| | \/ (-32*im(a*y) + 96*im(y) + 100*im(a) - 32*im(y)*re(y) - 28*im(a)*re(a)) + \-142 - 32*re(a*y) - 16*re (y) - 14*re (a) + 14*im (a) + 16*im (y) + 96*re(y) + 100*re(a)/ *cos|---------------------------------------------------------------------------------------------------------------------------------------------------------------------|
| \ 2 / | \ 2 /
x1 = I*|- ------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------ + im(a)| - ------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------ + re(a)
\ 4 / 4
$$x_{1} = i \left(- \frac{\sqrt[4]{\left(- 28 \operatorname{re}{\left(a\right)} \operatorname{im}{\left(a\right)} - 32 \operatorname{re}{\left(y\right)} \operatorname{im}{\left(y\right)} + 100 \operatorname{im}{\left(a\right)} + 96 \operatorname{im}{\left(y\right)} - 32 \operatorname{im}{\left(a y\right)}\right)^{2} + \left(- 14 \left(\operatorname{re}{\left(a\right)}\right)^{2} + 100 \operatorname{re}{\left(a\right)} - 16 \left(\operatorname{re}{\left(y\right)}\right)^{2} + 96 \operatorname{re}{\left(y\right)} - 32 \operatorname{re}{\left(a y\right)} + 14 \left(\operatorname{im}{\left(a\right)}\right)^{2} + 16 \left(\operatorname{im}{\left(y\right)}\right)^{2} - 142\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(- 28 \operatorname{re}{\left(a\right)} \operatorname{im}{\left(a\right)} - 32 \operatorname{re}{\left(y\right)} \operatorname{im}{\left(y\right)} + 100 \operatorname{im}{\left(a\right)} + 96 \operatorname{im}{\left(y\right)} - 32 \operatorname{im}{\left(a y\right)},- 14 \left(\operatorname{re}{\left(a\right)}\right)^{2} + 100 \operatorname{re}{\left(a\right)} - 16 \left(\operatorname{re}{\left(y\right)}\right)^{2} + 96 \operatorname{re}{\left(y\right)} - 32 \operatorname{re}{\left(a y\right)} + 14 \left(\operatorname{im}{\left(a\right)}\right)^{2} + 16 \left(\operatorname{im}{\left(y\right)}\right)^{2} - 142 \right)}}{2} \right)}}{4} + \operatorname{im}{\left(a\right)}\right) - \frac{\sqrt[4]{\left(- 28 \operatorname{re}{\left(a\right)} \operatorname{im}{\left(a\right)} - 32 \operatorname{re}{\left(y\right)} \operatorname{im}{\left(y\right)} + 100 \operatorname{im}{\left(a\right)} + 96 \operatorname{im}{\left(y\right)} - 32 \operatorname{im}{\left(a y\right)}\right)^{2} + \left(- 14 \left(\operatorname{re}{\left(a\right)}\right)^{2} + 100 \operatorname{re}{\left(a\right)} - 16 \left(\operatorname{re}{\left(y\right)}\right)^{2} + 96 \operatorname{re}{\left(y\right)} - 32 \operatorname{re}{\left(a y\right)} + 14 \left(\operatorname{im}{\left(a\right)}\right)^{2} + 16 \left(\operatorname{im}{\left(y\right)}\right)^{2} - 142\right)^{2}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(- 28 \operatorname{re}{\left(a\right)} \operatorname{im}{\left(a\right)} - 32 \operatorname{re}{\left(y\right)} \operatorname{im}{\left(y\right)} + 100 \operatorname{im}{\left(a\right)} + 96 \operatorname{im}{\left(y\right)} - 32 \operatorname{im}{\left(a y\right)},- 14 \left(\operatorname{re}{\left(a\right)}\right)^{2} + 100 \operatorname{re}{\left(a\right)} - 16 \left(\operatorname{re}{\left(y\right)}\right)^{2} + 96 \operatorname{re}{\left(y\right)} - 32 \operatorname{re}{\left(a y\right)} + 14 \left(\operatorname{im}{\left(a\right)}\right)^{2} + 16 \left(\operatorname{im}{\left(y\right)}\right)^{2} - 142 \right)}}{2} \right)}}{4} + \operatorname{re}{\left(a\right)}$$
/ _______________________________________________________________________________________________________________________________________________________________________ \ _______________________________________________________________________________________________________________________________________________________________________
| / 2 / / 2 2 2 2 \\ | / 2 / / 2 2 2 2 \\
|4 / 2 / 2 2 2 2 \ |atan2\-32*im(a*y) + 96*im(y) + 100*im(a) - 32*im(y)*re(y) - 28*im(a)*re(a), -142 - 32*re(a*y) - 16*re (y) - 14*re (a) + 14*im (a) + 16*im (y) + 96*re(y) + 100*re(a)/| | 4 / 2 / 2 2 2 2 \ |atan2\-32*im(a*y) + 96*im(y) + 100*im(a) - 32*im(y)*re(y) - 28*im(a)*re(a), -142 - 32*re(a*y) - 16*re (y) - 14*re (a) + 14*im (a) + 16*im (y) + 96*re(y) + 100*re(a)/|
|\/ (-32*im(a*y) + 96*im(y) + 100*im(a) - 32*im(y)*re(y) - 28*im(a)*re(a)) + \-142 - 32*re(a*y) - 16*re (y) - 14*re (a) + 14*im (a) + 16*im (y) + 96*re(y) + 100*re(a)/ *sin|---------------------------------------------------------------------------------------------------------------------------------------------------------------------| | \/ (-32*im(a*y) + 96*im(y) + 100*im(a) - 32*im(y)*re(y) - 28*im(a)*re(a)) + \-142 - 32*re(a*y) - 16*re (y) - 14*re (a) + 14*im (a) + 16*im (y) + 96*re(y) + 100*re(a)/ *cos|---------------------------------------------------------------------------------------------------------------------------------------------------------------------|
| \ 2 / | \ 2 /
x2 = I*|------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------ + im(a)| + ------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------ + re(a)
\ 4 / 4
$$x_{2} = i \left(\frac{\sqrt[4]{\left(- 28 \operatorname{re}{\left(a\right)} \operatorname{im}{\left(a\right)} - 32 \operatorname{re}{\left(y\right)} \operatorname{im}{\left(y\right)} + 100 \operatorname{im}{\left(a\right)} + 96 \operatorname{im}{\left(y\right)} - 32 \operatorname{im}{\left(a y\right)}\right)^{2} + \left(- 14 \left(\operatorname{re}{\left(a\right)}\right)^{2} + 100 \operatorname{re}{\left(a\right)} - 16 \left(\operatorname{re}{\left(y\right)}\right)^{2} + 96 \operatorname{re}{\left(y\right)} - 32 \operatorname{re}{\left(a y\right)} + 14 \left(\operatorname{im}{\left(a\right)}\right)^{2} + 16 \left(\operatorname{im}{\left(y\right)}\right)^{2} - 142\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(- 28 \operatorname{re}{\left(a\right)} \operatorname{im}{\left(a\right)} - 32 \operatorname{re}{\left(y\right)} \operatorname{im}{\left(y\right)} + 100 \operatorname{im}{\left(a\right)} + 96 \operatorname{im}{\left(y\right)} - 32 \operatorname{im}{\left(a y\right)},- 14 \left(\operatorname{re}{\left(a\right)}\right)^{2} + 100 \operatorname{re}{\left(a\right)} - 16 \left(\operatorname{re}{\left(y\right)}\right)^{2} + 96 \operatorname{re}{\left(y\right)} - 32 \operatorname{re}{\left(a y\right)} + 14 \left(\operatorname{im}{\left(a\right)}\right)^{2} + 16 \left(\operatorname{im}{\left(y\right)}\right)^{2} - 142 \right)}}{2} \right)}}{4} + \operatorname{im}{\left(a\right)}\right) + \frac{\sqrt[4]{\left(- 28 \operatorname{re}{\left(a\right)} \operatorname{im}{\left(a\right)} - 32 \operatorname{re}{\left(y\right)} \operatorname{im}{\left(y\right)} + 100 \operatorname{im}{\left(a\right)} + 96 \operatorname{im}{\left(y\right)} - 32 \operatorname{im}{\left(a y\right)}\right)^{2} + \left(- 14 \left(\operatorname{re}{\left(a\right)}\right)^{2} + 100 \operatorname{re}{\left(a\right)} - 16 \left(\operatorname{re}{\left(y\right)}\right)^{2} + 96 \operatorname{re}{\left(y\right)} - 32 \operatorname{re}{\left(a y\right)} + 14 \left(\operatorname{im}{\left(a\right)}\right)^{2} + 16 \left(\operatorname{im}{\left(y\right)}\right)^{2} - 142\right)^{2}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(- 28 \operatorname{re}{\left(a\right)} \operatorname{im}{\left(a\right)} - 32 \operatorname{re}{\left(y\right)} \operatorname{im}{\left(y\right)} + 100 \operatorname{im}{\left(a\right)} + 96 \operatorname{im}{\left(y\right)} - 32 \operatorname{im}{\left(a y\right)},- 14 \left(\operatorname{re}{\left(a\right)}\right)^{2} + 100 \operatorname{re}{\left(a\right)} - 16 \left(\operatorname{re}{\left(y\right)}\right)^{2} + 96 \operatorname{re}{\left(y\right)} - 32 \operatorname{re}{\left(a y\right)} + 14 \left(\operatorname{im}{\left(a\right)}\right)^{2} + 16 \left(\operatorname{im}{\left(y\right)}\right)^{2} - 142 \right)}}{2} \right)}}{4} + \operatorname{re}{\left(a\right)}$$
x2 = i*(((-28*re(a)*im(a) - 32*re(y)*im(y) + 100*im(a) + 96*im(y) - 32*im(a*y))^2 + (-14*re(a)^2 + 100*re(a) - 16*re(y)^2 + 96*re(y) - 32*re(a*y) + 14*im(a)^2 + 16*im(y)^2 - 142)^2)^(1/4)*sin(atan2(-28*re(a)*im(a) - 32*re(y)*im(y) + 100*im(a) + 96*im(y) - 32*im(a*y, -14*re(a)^2 + 100*re(a) - 16*re(y)^2 + 96*re(y) - 32*re(a*y) + 14*im(a)^2 + 16*im(y)^2 - 142)/2)/4 + im(a)) + ((-28*re(a)*im(a) - 32*re(y)*im(y) + 100*im(a) + 96*im(y) - 32*im(a*y))^2 + (-14*re(a)^2 + 100*re(a) - 16*re(y)^2 + 96*re(y) - 32*re(a*y) + 14*im(a)^2 + 16*im(y)^2 - 142)^2)^(1/4)*cos(atan2(-28*re(a)*im(a) - 32*re(y)*im(y) + 100*im(a) + 96*im(y) - 32*im(a*y), -14*re(a)^2 + 100*re(a) - 16*re(y)^2 + 96*re(y) - 32*re(a*y) + 14*im(a)^2 + 16*im(y)^2 - 142)/2)/4 + re(a))
Sum and product of roots
[src]
sum
/ _______________________________________________________________________________________________________________________________________________________________________ \ _______________________________________________________________________________________________________________________________________________________________________ / _______________________________________________________________________________________________________________________________________________________________________ \ _______________________________________________________________________________________________________________________________________________________________________ | / 2 / / 2 2 2 2 \\ | / 2 / / 2 2 2 2 \\ | / 2 / / 2 2 2 2 \\ | / 2 / / 2 2 2 2 \\ | 4 / 2 / 2 2 2 2 \ |atan2\-32*im(a*y) + 96*im(y) + 100*im(a) - 32*im(y)*re(y) - 28*im(a)*re(a), -142 - 32*re(a*y) - 16*re (y) - 14*re (a) + 14*im (a) + 16*im (y) + 96*re(y) + 100*re(a)/| | 4 / 2 / 2 2 2 2 \ |atan2\-32*im(a*y) + 96*im(y) + 100*im(a) - 32*im(y)*re(y) - 28*im(a)*re(a), -142 - 32*re(a*y) - 16*re (y) - 14*re (a) + 14*im (a) + 16*im (y) + 96*re(y) + 100*re(a)/| |4 / 2 / 2 2 2 2 \ |atan2\-32*im(a*y) + 96*im(y) + 100*im(a) - 32*im(y)*re(y) - 28*im(a)*re(a), -142 - 32*re(a*y) - 16*re (y) - 14*re (a) + 14*im (a) + 16*im (y) + 96*re(y) + 100*re(a)/| | 4 / 2 / 2 2 2 2 \ |atan2\-32*im(a*y) + 96*im(y) + 100*im(a) - 32*im(y)*re(y) - 28*im(a)*re(a), -142 - 32*re(a*y) - 16*re (y) - 14*re (a) + 14*im (a) + 16*im (y) + 96*re(y) + 100*re(a)/| | \/ (-32*im(a*y) + 96*im(y) + 100*im(a) - 32*im(y)*re(y) - 28*im(a)*re(a)) + \-142 - 32*re(a*y) - 16*re (y) - 14*re (a) + 14*im (a) + 16*im (y) + 96*re(y) + 100*re(a)/ *sin|---------------------------------------------------------------------------------------------------------------------------------------------------------------------| | \/ (-32*im(a*y) + 96*im(y) + 100*im(a) - 32*im(y)*re(y) - 28*im(a)*re(a)) + \-142 - 32*re(a*y) - 16*re (y) - 14*re (a) + 14*im (a) + 16*im (y) + 96*re(y) + 100*re(a)/ *cos|---------------------------------------------------------------------------------------------------------------------------------------------------------------------| |\/ (-32*im(a*y) + 96*im(y) + 100*im(a) - 32*im(y)*re(y) - 28*im(a)*re(a)) + \-142 - 32*re(a*y) - 16*re (y) - 14*re (a) + 14*im (a) + 16*im (y) + 96*re(y) + 100*re(a)/ *sin|---------------------------------------------------------------------------------------------------------------------------------------------------------------------| | \/ (-32*im(a*y) + 96*im(y) + 100*im(a) - 32*im(y)*re(y) - 28*im(a)*re(a)) + \-142 - 32*re(a*y) - 16*re (y) - 14*re (a) + 14*im (a) + 16*im (y) + 96*re(y) + 100*re(a)/ *cos|---------------------------------------------------------------------------------------------------------------------------------------------------------------------| | \ 2 / | \ 2 / | \ 2 / | \ 2 / I*|- ------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------ + im(a)| - ------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------ + re(a) + I*|------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------ + im(a)| + ------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------ + re(a) \ 4 / 4 \ 4 / 4
$$\left(i \left(- \frac{\sqrt[4]{\left(- 28 \operatorname{re}{\left(a\right)} \operatorname{im}{\left(a\right)} - 32 \operatorname{re}{\left(y\right)} \operatorname{im}{\left(y\right)} + 100 \operatorname{im}{\left(a\right)} + 96 \operatorname{im}{\left(y\right)} - 32 \operatorname{im}{\left(a y\right)}\right)^{2} + \left(- 14 \left(\operatorname{re}{\left(a\right)}\right)^{2} + 100 \operatorname{re}{\left(a\right)} - 16 \left(\operatorname{re}{\left(y\right)}\right)^{2} + 96 \operatorname{re}{\left(y\right)} - 32 \operatorname{re}{\left(a y\right)} + 14 \left(\operatorname{im}{\left(a\right)}\right)^{2} + 16 \left(\operatorname{im}{\left(y\right)}\right)^{2} - 142\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(- 28 \operatorname{re}{\left(a\right)} \operatorname{im}{\left(a\right)} - 32 \operatorname{re}{\left(y\right)} \operatorname{im}{\left(y\right)} + 100 \operatorname{im}{\left(a\right)} + 96 \operatorname{im}{\left(y\right)} - 32 \operatorname{im}{\left(a y\right)},- 14 \left(\operatorname{re}{\left(a\right)}\right)^{2} + 100 \operatorname{re}{\left(a\right)} - 16 \left(\operatorname{re}{\left(y\right)}\right)^{2} + 96 \operatorname{re}{\left(y\right)} - 32 \operatorname{re}{\left(a y\right)} + 14 \left(\operatorname{im}{\left(a\right)}\right)^{2} + 16 \left(\operatorname{im}{\left(y\right)}\right)^{2} - 142 \right)}}{2} \right)}}{4} + \operatorname{im}{\left(a\right)}\right) - \frac{\sqrt[4]{\left(- 28 \operatorname{re}{\left(a\right)} \operatorname{im}{\left(a\right)} - 32 \operatorname{re}{\left(y\right)} \operatorname{im}{\left(y\right)} + 100 \operatorname{im}{\left(a\right)} + 96 \operatorname{im}{\left(y\right)} - 32 \operatorname{im}{\left(a y\right)}\right)^{2} + \left(- 14 \left(\operatorname{re}{\left(a\right)}\right)^{2} + 100 \operatorname{re}{\left(a\right)} - 16 \left(\operatorname{re}{\left(y\right)}\right)^{2} + 96 \operatorname{re}{\left(y\right)} - 32 \operatorname{re}{\left(a y\right)} + 14 \left(\operatorname{im}{\left(a\right)}\right)^{2} + 16 \left(\operatorname{im}{\left(y\right)}\right)^{2} - 142\right)^{2}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(- 28 \operatorname{re}{\left(a\right)} \operatorname{im}{\left(a\right)} - 32 \operatorname{re}{\left(y\right)} \operatorname{im}{\left(y\right)} + 100 \operatorname{im}{\left(a\right)} + 96 \operatorname{im}{\left(y\right)} - 32 \operatorname{im}{\left(a y\right)},- 14 \left(\operatorname{re}{\left(a\right)}\right)^{2} + 100 \operatorname{re}{\left(a\right)} - 16 \left(\operatorname{re}{\left(y\right)}\right)^{2} + 96 \operatorname{re}{\left(y\right)} - 32 \operatorname{re}{\left(a y\right)} + 14 \left(\operatorname{im}{\left(a\right)}\right)^{2} + 16 \left(\operatorname{im}{\left(y\right)}\right)^{2} - 142 \right)}}{2} \right)}}{4} + \operatorname{re}{\left(a\right)}\right) + \left(i \left(\frac{\sqrt[4]{\left(- 28 \operatorname{re}{\left(a\right)} \operatorname{im}{\left(a\right)} - 32 \operatorname{re}{\left(y\right)} \operatorname{im}{\left(y\right)} + 100 \operatorname{im}{\left(a\right)} + 96 \operatorname{im}{\left(y\right)} - 32 \operatorname{im}{\left(a y\right)}\right)^{2} + \left(- 14 \left(\operatorname{re}{\left(a\right)}\right)^{2} + 100 \operatorname{re}{\left(a\right)} - 16 \left(\operatorname{re}{\left(y\right)}\right)^{2} + 96 \operatorname{re}{\left(y\right)} - 32 \operatorname{re}{\left(a y\right)} + 14 \left(\operatorname{im}{\left(a\right)}\right)^{2} + 16 \left(\operatorname{im}{\left(y\right)}\right)^{2} - 142\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(- 28 \operatorname{re}{\left(a\right)} \operatorname{im}{\left(a\right)} - 32 \operatorname{re}{\left(y\right)} \operatorname{im}{\left(y\right)} + 100 \operatorname{im}{\left(a\right)} + 96 \operatorname{im}{\left(y\right)} - 32 \operatorname{im}{\left(a y\right)},- 14 \left(\operatorname{re}{\left(a\right)}\right)^{2} + 100 \operatorname{re}{\left(a\right)} - 16 \left(\operatorname{re}{\left(y\right)}\right)^{2} + 96 \operatorname{re}{\left(y\right)} - 32 \operatorname{re}{\left(a y\right)} + 14 \left(\operatorname{im}{\left(a\right)}\right)^{2} + 16 \left(\operatorname{im}{\left(y\right)}\right)^{2} - 142 \right)}}{2} \right)}}{4} + \operatorname{im}{\left(a\right)}\right) + \frac{\sqrt[4]{\left(- 28 \operatorname{re}{\left(a\right)} \operatorname{im}{\left(a\right)} - 32 \operatorname{re}{\left(y\right)} \operatorname{im}{\left(y\right)} + 100 \operatorname{im}{\left(a\right)} + 96 \operatorname{im}{\left(y\right)} - 32 \operatorname{im}{\left(a y\right)}\right)^{2} + \left(- 14 \left(\operatorname{re}{\left(a\right)}\right)^{2} + 100 \operatorname{re}{\left(a\right)} - 16 \left(\operatorname{re}{\left(y\right)}\right)^{2} + 96 \operatorname{re}{\left(y\right)} - 32 \operatorname{re}{\left(a y\right)} + 14 \left(\operatorname{im}{\left(a\right)}\right)^{2} + 16 \left(\operatorname{im}{\left(y\right)}\right)^{2} - 142\right)^{2}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(- 28 \operatorname{re}{\left(a\right)} \operatorname{im}{\left(a\right)} - 32 \operatorname{re}{\left(y\right)} \operatorname{im}{\left(y\right)} + 100 \operatorname{im}{\left(a\right)} + 96 \operatorname{im}{\left(y\right)} - 32 \operatorname{im}{\left(a y\right)},- 14 \left(\operatorname{re}{\left(a\right)}\right)^{2} + 100 \operatorname{re}{\left(a\right)} - 16 \left(\operatorname{re}{\left(y\right)}\right)^{2} + 96 \operatorname{re}{\left(y\right)} - 32 \operatorname{re}{\left(a y\right)} + 14 \left(\operatorname{im}{\left(a\right)}\right)^{2} + 16 \left(\operatorname{im}{\left(y\right)}\right)^{2} - 142 \right)}}{2} \right)}}{4} + \operatorname{re}{\left(a\right)}\right)$$
=
/ _______________________________________________________________________________________________________________________________________________________________________ \ / _______________________________________________________________________________________________________________________________________________________________________ \
| / 2 / / 2 2 2 2 \\ | | / 2 / / 2 2 2 2 \\ |
| 4 / 2 / 2 2 2 2 \ |atan2\-32*im(a*y) + 96*im(y) + 100*im(a) - 32*im(y)*re(y) - 28*im(a)*re(a), -142 - 32*re(a*y) - 16*re (y) - 14*re (a) + 14*im (a) + 16*im (y) + 96*re(y) + 100*re(a)/| | |4 / 2 / 2 2 2 2 \ |atan2\-32*im(a*y) + 96*im(y) + 100*im(a) - 32*im(y)*re(y) - 28*im(a)*re(a), -142 - 32*re(a*y) - 16*re (y) - 14*re (a) + 14*im (a) + 16*im (y) + 96*re(y) + 100*re(a)/| |
| \/ (-32*im(a*y) + 96*im(y) + 100*im(a) - 32*im(y)*re(y) - 28*im(a)*re(a)) + \-142 - 32*re(a*y) - 16*re (y) - 14*re (a) + 14*im (a) + 16*im (y) + 96*re(y) + 100*re(a)/ *sin|---------------------------------------------------------------------------------------------------------------------------------------------------------------------| | |\/ (-32*im(a*y) + 96*im(y) + 100*im(a) - 32*im(y)*re(y) - 28*im(a)*re(a)) + \-142 - 32*re(a*y) - 16*re (y) - 14*re (a) + 14*im (a) + 16*im (y) + 96*re(y) + 100*re(a)/ *sin|---------------------------------------------------------------------------------------------------------------------------------------------------------------------| |
| \ 2 / | | \ 2 / |
2*re(a) + I*|- ------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------ + im(a)| + I*|------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------ + im(a)|
\ 4 / \ 4 /
$$i \left(- \frac{\sqrt[4]{\left(- 28 \operatorname{re}{\left(a\right)} \operatorname{im}{\left(a\right)} - 32 \operatorname{re}{\left(y\right)} \operatorname{im}{\left(y\right)} + 100 \operatorname{im}{\left(a\right)} + 96 \operatorname{im}{\left(y\right)} - 32 \operatorname{im}{\left(a y\right)}\right)^{2} + \left(- 14 \left(\operatorname{re}{\left(a\right)}\right)^{2} + 100 \operatorname{re}{\left(a\right)} - 16 \left(\operatorname{re}{\left(y\right)}\right)^{2} + 96 \operatorname{re}{\left(y\right)} - 32 \operatorname{re}{\left(a y\right)} + 14 \left(\operatorname{im}{\left(a\right)}\right)^{2} + 16 \left(\operatorname{im}{\left(y\right)}\right)^{2} - 142\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(- 28 \operatorname{re}{\left(a\right)} \operatorname{im}{\left(a\right)} - 32 \operatorname{re}{\left(y\right)} \operatorname{im}{\left(y\right)} + 100 \operatorname{im}{\left(a\right)} + 96 \operatorname{im}{\left(y\right)} - 32 \operatorname{im}{\left(a y\right)},- 14 \left(\operatorname{re}{\left(a\right)}\right)^{2} + 100 \operatorname{re}{\left(a\right)} - 16 \left(\operatorname{re}{\left(y\right)}\right)^{2} + 96 \operatorname{re}{\left(y\right)} - 32 \operatorname{re}{\left(a y\right)} + 14 \left(\operatorname{im}{\left(a\right)}\right)^{2} + 16 \left(\operatorname{im}{\left(y\right)}\right)^{2} - 142 \right)}}{2} \right)}}{4} + \operatorname{im}{\left(a\right)}\right) + i \left(\frac{\sqrt[4]{\left(- 28 \operatorname{re}{\left(a\right)} \operatorname{im}{\left(a\right)} - 32 \operatorname{re}{\left(y\right)} \operatorname{im}{\left(y\right)} + 100 \operatorname{im}{\left(a\right)} + 96 \operatorname{im}{\left(y\right)} - 32 \operatorname{im}{\left(a y\right)}\right)^{2} + \left(- 14 \left(\operatorname{re}{\left(a\right)}\right)^{2} + 100 \operatorname{re}{\left(a\right)} - 16 \left(\operatorname{re}{\left(y\right)}\right)^{2} + 96 \operatorname{re}{\left(y\right)} - 32 \operatorname{re}{\left(a y\right)} + 14 \left(\operatorname{im}{\left(a\right)}\right)^{2} + 16 \left(\operatorname{im}{\left(y\right)}\right)^{2} - 142\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(- 28 \operatorname{re}{\left(a\right)} \operatorname{im}{\left(a\right)} - 32 \operatorname{re}{\left(y\right)} \operatorname{im}{\left(y\right)} + 100 \operatorname{im}{\left(a\right)} + 96 \operatorname{im}{\left(y\right)} - 32 \operatorname{im}{\left(a y\right)},- 14 \left(\operatorname{re}{\left(a\right)}\right)^{2} + 100 \operatorname{re}{\left(a\right)} - 16 \left(\operatorname{re}{\left(y\right)}\right)^{2} + 96 \operatorname{re}{\left(y\right)} - 32 \operatorname{re}{\left(a y\right)} + 14 \left(\operatorname{im}{\left(a\right)}\right)^{2} + 16 \left(\operatorname{im}{\left(y\right)}\right)^{2} - 142 \right)}}{2} \right)}}{4} + \operatorname{im}{\left(a\right)}\right) + 2 \operatorname{re}{\left(a\right)}$$
product
/ / _______________________________________________________________________________________________________________________________________________________________________ \ _______________________________________________________________________________________________________________________________________________________________________ \ / / _______________________________________________________________________________________________________________________________________________________________________ \ _______________________________________________________________________________________________________________________________________________________________________ \ | | / 2 / / 2 2 2 2 \\ | / 2 / / 2 2 2 2 \\ | | | / 2 / / 2 2 2 2 \\ | / 2 / / 2 2 2 2 \\ | | | 4 / 2 / 2 2 2 2 \ |atan2\-32*im(a*y) + 96*im(y) + 100*im(a) - 32*im(y)*re(y) - 28*im(a)*re(a), -142 - 32*re(a*y) - 16*re (y) - 14*re (a) + 14*im (a) + 16*im (y) + 96*re(y) + 100*re(a)/| | 4 / 2 / 2 2 2 2 \ |atan2\-32*im(a*y) + 96*im(y) + 100*im(a) - 32*im(y)*re(y) - 28*im(a)*re(a), -142 - 32*re(a*y) - 16*re (y) - 14*re (a) + 14*im (a) + 16*im (y) + 96*re(y) + 100*re(a)/| | | |4 / 2 / 2 2 2 2 \ |atan2\-32*im(a*y) + 96*im(y) + 100*im(a) - 32*im(y)*re(y) - 28*im(a)*re(a), -142 - 32*re(a*y) - 16*re (y) - 14*re (a) + 14*im (a) + 16*im (y) + 96*re(y) + 100*re(a)/| | 4 / 2 / 2 2 2 2 \ |atan2\-32*im(a*y) + 96*im(y) + 100*im(a) - 32*im(y)*re(y) - 28*im(a)*re(a), -142 - 32*re(a*y) - 16*re (y) - 14*re (a) + 14*im (a) + 16*im (y) + 96*re(y) + 100*re(a)/| | | | \/ (-32*im(a*y) + 96*im(y) + 100*im(a) - 32*im(y)*re(y) - 28*im(a)*re(a)) + \-142 - 32*re(a*y) - 16*re (y) - 14*re (a) + 14*im (a) + 16*im (y) + 96*re(y) + 100*re(a)/ *sin|---------------------------------------------------------------------------------------------------------------------------------------------------------------------| | \/ (-32*im(a*y) + 96*im(y) + 100*im(a) - 32*im(y)*re(y) - 28*im(a)*re(a)) + \-142 - 32*re(a*y) - 16*re (y) - 14*re (a) + 14*im (a) + 16*im (y) + 96*re(y) + 100*re(a)/ *cos|---------------------------------------------------------------------------------------------------------------------------------------------------------------------| | | |\/ (-32*im(a*y) + 96*im(y) + 100*im(a) - 32*im(y)*re(y) - 28*im(a)*re(a)) + \-142 - 32*re(a*y) - 16*re (y) - 14*re (a) + 14*im (a) + 16*im (y) + 96*re(y) + 100*re(a)/ *sin|---------------------------------------------------------------------------------------------------------------------------------------------------------------------| | \/ (-32*im(a*y) + 96*im(y) + 100*im(a) - 32*im(y)*re(y) - 28*im(a)*re(a)) + \-142 - 32*re(a*y) - 16*re (y) - 14*re (a) + 14*im (a) + 16*im (y) + 96*re(y) + 100*re(a)/ *cos|---------------------------------------------------------------------------------------------------------------------------------------------------------------------| | | | \ 2 / | \ 2 / | | | \ 2 / | \ 2 / | |I*|- ------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------ + im(a)| - ------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------ + re(a)|*|I*|------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------ + im(a)| + ------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------ + re(a)| \ \ 4 / 4 / \ \ 4 / 4 /
$$\left(i \left(- \frac{\sqrt[4]{\left(- 28 \operatorname{re}{\left(a\right)} \operatorname{im}{\left(a\right)} - 32 \operatorname{re}{\left(y\right)} \operatorname{im}{\left(y\right)} + 100 \operatorname{im}{\left(a\right)} + 96 \operatorname{im}{\left(y\right)} - 32 \operatorname{im}{\left(a y\right)}\right)^{2} + \left(- 14 \left(\operatorname{re}{\left(a\right)}\right)^{2} + 100 \operatorname{re}{\left(a\right)} - 16 \left(\operatorname{re}{\left(y\right)}\right)^{2} + 96 \operatorname{re}{\left(y\right)} - 32 \operatorname{re}{\left(a y\right)} + 14 \left(\operatorname{im}{\left(a\right)}\right)^{2} + 16 \left(\operatorname{im}{\left(y\right)}\right)^{2} - 142\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(- 28 \operatorname{re}{\left(a\right)} \operatorname{im}{\left(a\right)} - 32 \operatorname{re}{\left(y\right)} \operatorname{im}{\left(y\right)} + 100 \operatorname{im}{\left(a\right)} + 96 \operatorname{im}{\left(y\right)} - 32 \operatorname{im}{\left(a y\right)},- 14 \left(\operatorname{re}{\left(a\right)}\right)^{2} + 100 \operatorname{re}{\left(a\right)} - 16 \left(\operatorname{re}{\left(y\right)}\right)^{2} + 96 \operatorname{re}{\left(y\right)} - 32 \operatorname{re}{\left(a y\right)} + 14 \left(\operatorname{im}{\left(a\right)}\right)^{2} + 16 \left(\operatorname{im}{\left(y\right)}\right)^{2} - 142 \right)}}{2} \right)}}{4} + \operatorname{im}{\left(a\right)}\right) - \frac{\sqrt[4]{\left(- 28 \operatorname{re}{\left(a\right)} \operatorname{im}{\left(a\right)} - 32 \operatorname{re}{\left(y\right)} \operatorname{im}{\left(y\right)} + 100 \operatorname{im}{\left(a\right)} + 96 \operatorname{im}{\left(y\right)} - 32 \operatorname{im}{\left(a y\right)}\right)^{2} + \left(- 14 \left(\operatorname{re}{\left(a\right)}\right)^{2} + 100 \operatorname{re}{\left(a\right)} - 16 \left(\operatorname{re}{\left(y\right)}\right)^{2} + 96 \operatorname{re}{\left(y\right)} - 32 \operatorname{re}{\left(a y\right)} + 14 \left(\operatorname{im}{\left(a\right)}\right)^{2} + 16 \left(\operatorname{im}{\left(y\right)}\right)^{2} - 142\right)^{2}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(- 28 \operatorname{re}{\left(a\right)} \operatorname{im}{\left(a\right)} - 32 \operatorname{re}{\left(y\right)} \operatorname{im}{\left(y\right)} + 100 \operatorname{im}{\left(a\right)} + 96 \operatorname{im}{\left(y\right)} - 32 \operatorname{im}{\left(a y\right)},- 14 \left(\operatorname{re}{\left(a\right)}\right)^{2} + 100 \operatorname{re}{\left(a\right)} - 16 \left(\operatorname{re}{\left(y\right)}\right)^{2} + 96 \operatorname{re}{\left(y\right)} - 32 \operatorname{re}{\left(a y\right)} + 14 \left(\operatorname{im}{\left(a\right)}\right)^{2} + 16 \left(\operatorname{im}{\left(y\right)}\right)^{2} - 142 \right)}}{2} \right)}}{4} + \operatorname{re}{\left(a\right)}\right) \left(i \left(\frac{\sqrt[4]{\left(- 28 \operatorname{re}{\left(a\right)} \operatorname{im}{\left(a\right)} - 32 \operatorname{re}{\left(y\right)} \operatorname{im}{\left(y\right)} + 100 \operatorname{im}{\left(a\right)} + 96 \operatorname{im}{\left(y\right)} - 32 \operatorname{im}{\left(a y\right)}\right)^{2} + \left(- 14 \left(\operatorname{re}{\left(a\right)}\right)^{2} + 100 \operatorname{re}{\left(a\right)} - 16 \left(\operatorname{re}{\left(y\right)}\right)^{2} + 96 \operatorname{re}{\left(y\right)} - 32 \operatorname{re}{\left(a y\right)} + 14 \left(\operatorname{im}{\left(a\right)}\right)^{2} + 16 \left(\operatorname{im}{\left(y\right)}\right)^{2} - 142\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(- 28 \operatorname{re}{\left(a\right)} \operatorname{im}{\left(a\right)} - 32 \operatorname{re}{\left(y\right)} \operatorname{im}{\left(y\right)} + 100 \operatorname{im}{\left(a\right)} + 96 \operatorname{im}{\left(y\right)} - 32 \operatorname{im}{\left(a y\right)},- 14 \left(\operatorname{re}{\left(a\right)}\right)^{2} + 100 \operatorname{re}{\left(a\right)} - 16 \left(\operatorname{re}{\left(y\right)}\right)^{2} + 96 \operatorname{re}{\left(y\right)} - 32 \operatorname{re}{\left(a y\right)} + 14 \left(\operatorname{im}{\left(a\right)}\right)^{2} + 16 \left(\operatorname{im}{\left(y\right)}\right)^{2} - 142 \right)}}{2} \right)}}{4} + \operatorname{im}{\left(a\right)}\right) + \frac{\sqrt[4]{\left(- 28 \operatorname{re}{\left(a\right)} \operatorname{im}{\left(a\right)} - 32 \operatorname{re}{\left(y\right)} \operatorname{im}{\left(y\right)} + 100 \operatorname{im}{\left(a\right)} + 96 \operatorname{im}{\left(y\right)} - 32 \operatorname{im}{\left(a y\right)}\right)^{2} + \left(- 14 \left(\operatorname{re}{\left(a\right)}\right)^{2} + 100 \operatorname{re}{\left(a\right)} - 16 \left(\operatorname{re}{\left(y\right)}\right)^{2} + 96 \operatorname{re}{\left(y\right)} - 32 \operatorname{re}{\left(a y\right)} + 14 \left(\operatorname{im}{\left(a\right)}\right)^{2} + 16 \left(\operatorname{im}{\left(y\right)}\right)^{2} - 142\right)^{2}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(- 28 \operatorname{re}{\left(a\right)} \operatorname{im}{\left(a\right)} - 32 \operatorname{re}{\left(y\right)} \operatorname{im}{\left(y\right)} + 100 \operatorname{im}{\left(a\right)} + 96 \operatorname{im}{\left(y\right)} - 32 \operatorname{im}{\left(a y\right)},- 14 \left(\operatorname{re}{\left(a\right)}\right)^{2} + 100 \operatorname{re}{\left(a\right)} - 16 \left(\operatorname{re}{\left(y\right)}\right)^{2} + 96 \operatorname{re}{\left(y\right)} - 32 \operatorname{re}{\left(a y\right)} + 14 \left(\operatorname{im}{\left(a\right)}\right)^{2} + 16 \left(\operatorname{im}{\left(y\right)}\right)^{2} - 142 \right)}}{2} \right)}}{4} + \operatorname{re}{\left(a\right)}\right)$$
=
2 2 71 2 2 25*re(a) 15*im (a) 15*re (a) 25*I*im(a) 15*I*im(a)*re(a) -- + re (y) - im (y) - 6*re(y) + 2*re(a*y) - -------- - --------- + --------- - 6*I*im(y) + 2*I*im(a*y) - ---------- + 2*I*im(y)*re(y) + ---------------- 8 4 8 8 4 4
$$\frac{15 \left(\operatorname{re}{\left(a\right)}\right)^{2}}{8} + \frac{15 i \operatorname{re}{\left(a\right)} \operatorname{im}{\left(a\right)}}{4} - \frac{25 \operatorname{re}{\left(a\right)}}{4} + \left(\operatorname{re}{\left(y\right)}\right)^{2} + 2 i \operatorname{re}{\left(y\right)} \operatorname{im}{\left(y\right)} - 6 \operatorname{re}{\left(y\right)} + 2 \operatorname{re}{\left(a y\right)} - \frac{15 \left(\operatorname{im}{\left(a\right)}\right)^{2}}{8} - \frac{25 i \operatorname{im}{\left(a\right)}}{4} - \left(\operatorname{im}{\left(y\right)}\right)^{2} - 6 i \operatorname{im}{\left(y\right)} + 2 i \operatorname{im}{\left(a y\right)} + \frac{71}{8}$$
71/8 + re(y)^2 - im(y)^2 - 6*re(y) + 2*re(a*y) - 25*re(a)/4 - 15*im(a)^2/8 + 15*re(a)^2/8 - 6*i*im(y) + 2*i*im(a*y) - 25*i*im(a)/4 + 2*i*im(y)*re(y) + 15*i*im(a)*re(a)/4