Mister Exam
x+y=(x-y)^2-12 equation
The teacher will be very surprised to see your correct solution 😉
The solution
Detail solution
Move right part of the equation to
left part with negative sign.
The equation is transformed from
$$x + y = \left(x - y\right)^{2} - 12$$
to
$$\left(12 - \left(x - y\right)^{2}\right) + \left(x + y\right) = 0$$
Expand the expression in the equation
$$\left(12 - \left(x - y\right)^{2}\right) + \left(x + y\right) = 0$$
We get the quadratic equation
$$- x^{2} + 2 x y + x - y^{2} + y + 12 = 0$$
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
$$a = -1$$
$$b = 2 y + 1$$
$$c = - y^{2} + y + 12$$
, then
The equation has two roots.
or
$$x_{1} = y - \frac{\sqrt{- 4 y^{2} + 4 y + \left(2 y + 1\right)^{2} + 48}}{2} + \frac{1}{2}$$
$$x_{2} = y + \frac{\sqrt{- 4 y^{2} + 4 y + \left(2 y + 1\right)^{2} + 48}}{2} + \frac{1}{2}$$
left part with negative sign.
The equation is transformed from
$$x + y = \left(x - y\right)^{2} - 12$$
to
$$\left(12 - \left(x - y\right)^{2}\right) + \left(x + y\right) = 0$$
Expand the expression in the equation
$$\left(12 - \left(x - y\right)^{2}\right) + \left(x + y\right) = 0$$
We get the quadratic equation
$$- x^{2} + 2 x y + x - y^{2} + y + 12 = 0$$
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
$$a = -1$$
$$b = 2 y + 1$$
$$c = - y^{2} + y + 12$$
, then
D = b^2 - 4 * a * c =
(1 + 2*y)^2 - 4 * (-1) * (12 + y - y^2) = 48 + (1 + 2*y)^2 - 4*y^2 + 4*y
The equation has two roots.
x1 = (-b + sqrt(D)) / (2*a)
x2 = (-b - sqrt(D)) / (2*a)
or
$$x_{1} = y - \frac{\sqrt{- 4 y^{2} + 4 y + \left(2 y + 1\right)^{2} + 48}}{2} + \frac{1}{2}$$
$$x_{2} = y + \frac{\sqrt{- 4 y^{2} + 4 y + \left(2 y + 1\right)^{2} + 48}}{2} + \frac{1}{2}$$
The graph
Sum and product of roots
[src]
sum
/ _____________________________ \ _____________________________ / _____________________________ \ _____________________________
| 4 / 2 2 /atan2(8*im(y), 49 + 8*re(y))\ | 4 / 2 2 /atan2(8*im(y), 49 + 8*re(y))\ |4 / 2 2 /atan2(8*im(y), 49 + 8*re(y))\ | 4 / 2 2 /atan2(8*im(y), 49 + 8*re(y))\
| \/ (49 + 8*re(y)) + 64*im (y) *sin|----------------------------| | \/ (49 + 8*re(y)) + 64*im (y) *cos|----------------------------| |\/ (49 + 8*re(y)) + 64*im (y) *sin|----------------------------| | \/ (49 + 8*re(y)) + 64*im (y) *cos|----------------------------|
1 | \ 2 / | \ 2 / 1 | \ 2 / | \ 2 /
- + I*|- ------------------------------------------------------------------ + im(y)| - ------------------------------------------------------------------ + re(y) + - + I*|------------------------------------------------------------------ + im(y)| + ------------------------------------------------------------------ + re(y)
2 \ 2 / 2 2 \ 2 / 2
$$\left(i \left(- \frac{\sqrt[4]{\left(8 \operatorname{re}{\left(y\right)} + 49\right)^{2} + 64 \left(\operatorname{im}{\left(y\right)}\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(8 \operatorname{im}{\left(y\right)},8 \operatorname{re}{\left(y\right)} + 49 \right)}}{2} \right)}}{2} + \operatorname{im}{\left(y\right)}\right) - \frac{\sqrt[4]{\left(8 \operatorname{re}{\left(y\right)} + 49\right)^{2} + 64 \left(\operatorname{im}{\left(y\right)}\right)^{2}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(8 \operatorname{im}{\left(y\right)},8 \operatorname{re}{\left(y\right)} + 49 \right)}}{2} \right)}}{2} + \operatorname{re}{\left(y\right)} + \frac{1}{2}\right) + \left(i \left(\frac{\sqrt[4]{\left(8 \operatorname{re}{\left(y\right)} + 49\right)^{2} + 64 \left(\operatorname{im}{\left(y\right)}\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(8 \operatorname{im}{\left(y\right)},8 \operatorname{re}{\left(y\right)} + 49 \right)}}{2} \right)}}{2} + \operatorname{im}{\left(y\right)}\right) + \frac{\sqrt[4]{\left(8 \operatorname{re}{\left(y\right)} + 49\right)^{2} + 64 \left(\operatorname{im}{\left(y\right)}\right)^{2}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(8 \operatorname{im}{\left(y\right)},8 \operatorname{re}{\left(y\right)} + 49 \right)}}{2} \right)}}{2} + \operatorname{re}{\left(y\right)} + \frac{1}{2}\right)$$
=
/ _____________________________ \ / _____________________________ \
|4 / 2 2 /atan2(8*im(y), 49 + 8*re(y))\ | | 4 / 2 2 /atan2(8*im(y), 49 + 8*re(y))\ |
|\/ (49 + 8*re(y)) + 64*im (y) *sin|----------------------------| | | \/ (49 + 8*re(y)) + 64*im (y) *sin|----------------------------| |
| \ 2 / | | \ 2 / |
1 + 2*re(y) + I*|------------------------------------------------------------------ + im(y)| + I*|- ------------------------------------------------------------------ + im(y)|
\ 2 / \ 2 /
$$i \left(- \frac{\sqrt[4]{\left(8 \operatorname{re}{\left(y\right)} + 49\right)^{2} + 64 \left(\operatorname{im}{\left(y\right)}\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(8 \operatorname{im}{\left(y\right)},8 \operatorname{re}{\left(y\right)} + 49 \right)}}{2} \right)}}{2} + \operatorname{im}{\left(y\right)}\right) + i \left(\frac{\sqrt[4]{\left(8 \operatorname{re}{\left(y\right)} + 49\right)^{2} + 64 \left(\operatorname{im}{\left(y\right)}\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(8 \operatorname{im}{\left(y\right)},8 \operatorname{re}{\left(y\right)} + 49 \right)}}{2} \right)}}{2} + \operatorname{im}{\left(y\right)}\right) + 2 \operatorname{re}{\left(y\right)} + 1$$
product
/ / _____________________________ \ _____________________________ \ / / _____________________________ \ _____________________________ \ | | 4 / 2 2 /atan2(8*im(y), 49 + 8*re(y))\ | 4 / 2 2 /atan2(8*im(y), 49 + 8*re(y))\ | | |4 / 2 2 /atan2(8*im(y), 49 + 8*re(y))\ | 4 / 2 2 /atan2(8*im(y), 49 + 8*re(y))\ | | | \/ (49 + 8*re(y)) + 64*im (y) *sin|----------------------------| | \/ (49 + 8*re(y)) + 64*im (y) *cos|----------------------------| | | |\/ (49 + 8*re(y)) + 64*im (y) *sin|----------------------------| | \/ (49 + 8*re(y)) + 64*im (y) *cos|----------------------------| | |1 | \ 2 / | \ 2 / | |1 | \ 2 / | \ 2 / | |- + I*|- ------------------------------------------------------------------ + im(y)| - ------------------------------------------------------------------ + re(y)|*|- + I*|------------------------------------------------------------------ + im(y)| + ------------------------------------------------------------------ + re(y)| \2 \ 2 / 2 / \2 \ 2 / 2 /
$$\left(i \left(- \frac{\sqrt[4]{\left(8 \operatorname{re}{\left(y\right)} + 49\right)^{2} + 64 \left(\operatorname{im}{\left(y\right)}\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(8 \operatorname{im}{\left(y\right)},8 \operatorname{re}{\left(y\right)} + 49 \right)}}{2} \right)}}{2} + \operatorname{im}{\left(y\right)}\right) - \frac{\sqrt[4]{\left(8 \operatorname{re}{\left(y\right)} + 49\right)^{2} + 64 \left(\operatorname{im}{\left(y\right)}\right)^{2}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(8 \operatorname{im}{\left(y\right)},8 \operatorname{re}{\left(y\right)} + 49 \right)}}{2} \right)}}{2} + \operatorname{re}{\left(y\right)} + \frac{1}{2}\right) \left(i \left(\frac{\sqrt[4]{\left(8 \operatorname{re}{\left(y\right)} + 49\right)^{2} + 64 \left(\operatorname{im}{\left(y\right)}\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(8 \operatorname{im}{\left(y\right)},8 \operatorname{re}{\left(y\right)} + 49 \right)}}{2} \right)}}{2} + \operatorname{im}{\left(y\right)}\right) + \frac{\sqrt[4]{\left(8 \operatorname{re}{\left(y\right)} + 49\right)^{2} + 64 \left(\operatorname{im}{\left(y\right)}\right)^{2}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(8 \operatorname{im}{\left(y\right)},8 \operatorname{re}{\left(y\right)} + 49 \right)}}{2} \right)}}{2} + \operatorname{re}{\left(y\right)} + \frac{1}{2}\right)$$
=
2 2 -12 + re (y) - im (y) - re(y) - I*im(y) + 2*I*im(y)*re(y)
$$\left(\operatorname{re}{\left(y\right)}\right)^{2} + 2 i \operatorname{re}{\left(y\right)} \operatorname{im}{\left(y\right)} - \operatorname{re}{\left(y\right)} - \left(\operatorname{im}{\left(y\right)}\right)^{2} - i \operatorname{im}{\left(y\right)} - 12$$
-12 + re(y)^2 - im(y)^2 - re(y) - i*im(y) + 2*i*im(y)*re(y)
Rapid solution
[src]
/ _____________________________ \ _____________________________
| 4 / 2 2 /atan2(8*im(y), 49 + 8*re(y))\ | 4 / 2 2 /atan2(8*im(y), 49 + 8*re(y))\
| \/ (49 + 8*re(y)) + 64*im (y) *sin|----------------------------| | \/ (49 + 8*re(y)) + 64*im (y) *cos|----------------------------|
1 | \ 2 / | \ 2 /
x1 = - + I*|- ------------------------------------------------------------------ + im(y)| - ------------------------------------------------------------------ + re(y)
2 \ 2 / 2
$$x_{1} = i \left(- \frac{\sqrt[4]{\left(8 \operatorname{re}{\left(y\right)} + 49\right)^{2} + 64 \left(\operatorname{im}{\left(y\right)}\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(8 \operatorname{im}{\left(y\right)},8 \operatorname{re}{\left(y\right)} + 49 \right)}}{2} \right)}}{2} + \operatorname{im}{\left(y\right)}\right) - \frac{\sqrt[4]{\left(8 \operatorname{re}{\left(y\right)} + 49\right)^{2} + 64 \left(\operatorname{im}{\left(y\right)}\right)^{2}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(8 \operatorname{im}{\left(y\right)},8 \operatorname{re}{\left(y\right)} + 49 \right)}}{2} \right)}}{2} + \operatorname{re}{\left(y\right)} + \frac{1}{2}$$
/ _____________________________ \ _____________________________
|4 / 2 2 /atan2(8*im(y), 49 + 8*re(y))\ | 4 / 2 2 /atan2(8*im(y), 49 + 8*re(y))\
|\/ (49 + 8*re(y)) + 64*im (y) *sin|----------------------------| | \/ (49 + 8*re(y)) + 64*im (y) *cos|----------------------------|
1 | \ 2 / | \ 2 /
x2 = - + I*|------------------------------------------------------------------ + im(y)| + ------------------------------------------------------------------ + re(y)
2 \ 2 / 2
$$x_{2} = i \left(\frac{\sqrt[4]{\left(8 \operatorname{re}{\left(y\right)} + 49\right)^{2} + 64 \left(\operatorname{im}{\left(y\right)}\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(8 \operatorname{im}{\left(y\right)},8 \operatorname{re}{\left(y\right)} + 49 \right)}}{2} \right)}}{2} + \operatorname{im}{\left(y\right)}\right) + \frac{\sqrt[4]{\left(8 \operatorname{re}{\left(y\right)} + 49\right)^{2} + 64 \left(\operatorname{im}{\left(y\right)}\right)^{2}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(8 \operatorname{im}{\left(y\right)},8 \operatorname{re}{\left(y\right)} + 49 \right)}}{2} \right)}}{2} + \operatorname{re}{\left(y\right)} + \frac{1}{2}$$
x2 = i*(((8*re(y) + 49)^2 + 64*im(y)^2)^(1/4)*sin(atan2(8*im(y, 8*re(y) + 49)/2)/2 + im(y)) + ((8*re(y) + 49)^2 + 64*im(y)^2)^(1/4)*cos(atan2(8*im(y), 8*re(y) + 49)/2)/2 + re(y) + 1/2)