Mister Exam
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Identical expressions
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Similar expressions
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- (x+3)^2+(y-3)^2=3
(x+3)^2+(y+3)^2=3 equation
The teacher will be very surprised to see your correct solution 😉
The solution
You have entered
[src]
2 2 (x + 3) + (y + 3) = 3
$$\left(x + 3\right)^{2} + \left(y + 3\right)^{2} = 3$$
Detail solution
Move right part of the equation to
left part with negative sign.
The equation is transformed from
$$\left(x + 3\right)^{2} + \left(y + 3\right)^{2} = 3$$
to
$$\left(\left(x + 3\right)^{2} + \left(y + 3\right)^{2}\right) - 3 = 0$$
Expand the expression in the equation
$$\left(\left(x + 3\right)^{2} + \left(y + 3\right)^{2}\right) - 3 = 0$$
We get the quadratic equation
$$x^{2} + 6 x + y^{2} + 6 y + 15 = 0$$
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
$$a = 1$$
$$b = 6$$
$$c = y^{2} + 6 y + 15$$
, then
The equation has two roots.
or
$$x_{1} = \frac{\sqrt{- 4 y^{2} - 24 y - 24}}{2} - 3$$
$$x_{2} = - \frac{\sqrt{- 4 y^{2} - 24 y - 24}}{2} - 3$$
left part with negative sign.
The equation is transformed from
$$\left(x + 3\right)^{2} + \left(y + 3\right)^{2} = 3$$
to
$$\left(\left(x + 3\right)^{2} + \left(y + 3\right)^{2}\right) - 3 = 0$$
Expand the expression in the equation
$$\left(\left(x + 3\right)^{2} + \left(y + 3\right)^{2}\right) - 3 = 0$$
We get the quadratic equation
$$x^{2} + 6 x + y^{2} + 6 y + 15 = 0$$
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
$$a = 1$$
$$b = 6$$
$$c = y^{2} + 6 y + 15$$
, then
D = b^2 - 4 * a * c =
(6)^2 - 4 * (1) * (15 + y^2 + 6*y) = -24 - 24*y - 4*y^2
The equation has two roots.
x1 = (-b + sqrt(D)) / (2*a)
x2 = (-b - sqrt(D)) / (2*a)
or
$$x_{1} = \frac{\sqrt{- 4 y^{2} - 24 y - 24}}{2} - 3$$
$$x_{2} = - \frac{\sqrt{- 4 y^{2} - 24 y - 24}}{2} - 3$$
The graph
Rapid solution
[src]
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/ 2 / / 2 2 \\ / 2 / / 2 2 \\
4 / 2 / 2 2 \ |atan2\-6*im(y) - 2*im(y)*re(y), -6 + im (y) - re (y) - 6*re(y)/| 4 / 2 / 2 2 \ |atan2\-6*im(y) - 2*im(y)*re(y), -6 + im (y) - re (y) - 6*re(y)/|
x1 = -3 - \/ (-6*im(y) - 2*im(y)*re(y)) + \-6 + im (y) - re (y) - 6*re(y)/ *cos|---------------------------------------------------------------| - I*\/ (-6*im(y) - 2*im(y)*re(y)) + \-6 + im (y) - re (y) - 6*re(y)/ *sin|---------------------------------------------------------------|
\ 2 / \ 2 /
$$x_{1} = - i \sqrt[4]{\left(- 2 \operatorname{re}{\left(y\right)} \operatorname{im}{\left(y\right)} - 6 \operatorname{im}{\left(y\right)}\right)^{2} + \left(- \left(\operatorname{re}{\left(y\right)}\right)^{2} - 6 \operatorname{re}{\left(y\right)} + \left(\operatorname{im}{\left(y\right)}\right)^{2} - 6\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(- 2 \operatorname{re}{\left(y\right)} \operatorname{im}{\left(y\right)} - 6 \operatorname{im}{\left(y\right)},- \left(\operatorname{re}{\left(y\right)}\right)^{2} - 6 \operatorname{re}{\left(y\right)} + \left(\operatorname{im}{\left(y\right)}\right)^{2} - 6 \right)}}{2} \right)} - \sqrt[4]{\left(- 2 \operatorname{re}{\left(y\right)} \operatorname{im}{\left(y\right)} - 6 \operatorname{im}{\left(y\right)}\right)^{2} + \left(- \left(\operatorname{re}{\left(y\right)}\right)^{2} - 6 \operatorname{re}{\left(y\right)} + \left(\operatorname{im}{\left(y\right)}\right)^{2} - 6\right)^{2}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(- 2 \operatorname{re}{\left(y\right)} \operatorname{im}{\left(y\right)} - 6 \operatorname{im}{\left(y\right)},- \left(\operatorname{re}{\left(y\right)}\right)^{2} - 6 \operatorname{re}{\left(y\right)} + \left(\operatorname{im}{\left(y\right)}\right)^{2} - 6 \right)}}{2} \right)} - 3$$
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/ 2 / / 2 2 \\ / 2 / / 2 2 \\
4 / 2 / 2 2 \ |atan2\-6*im(y) - 2*im(y)*re(y), -6 + im (y) - re (y) - 6*re(y)/| 4 / 2 / 2 2 \ |atan2\-6*im(y) - 2*im(y)*re(y), -6 + im (y) - re (y) - 6*re(y)/|
x2 = -3 + \/ (-6*im(y) - 2*im(y)*re(y)) + \-6 + im (y) - re (y) - 6*re(y)/ *cos|---------------------------------------------------------------| + I*\/ (-6*im(y) - 2*im(y)*re(y)) + \-6 + im (y) - re (y) - 6*re(y)/ *sin|---------------------------------------------------------------|
\ 2 / \ 2 /
$$x_{2} = i \sqrt[4]{\left(- 2 \operatorname{re}{\left(y\right)} \operatorname{im}{\left(y\right)} - 6 \operatorname{im}{\left(y\right)}\right)^{2} + \left(- \left(\operatorname{re}{\left(y\right)}\right)^{2} - 6 \operatorname{re}{\left(y\right)} + \left(\operatorname{im}{\left(y\right)}\right)^{2} - 6\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(- 2 \operatorname{re}{\left(y\right)} \operatorname{im}{\left(y\right)} - 6 \operatorname{im}{\left(y\right)},- \left(\operatorname{re}{\left(y\right)}\right)^{2} - 6 \operatorname{re}{\left(y\right)} + \left(\operatorname{im}{\left(y\right)}\right)^{2} - 6 \right)}}{2} \right)} + \sqrt[4]{\left(- 2 \operatorname{re}{\left(y\right)} \operatorname{im}{\left(y\right)} - 6 \operatorname{im}{\left(y\right)}\right)^{2} + \left(- \left(\operatorname{re}{\left(y\right)}\right)^{2} - 6 \operatorname{re}{\left(y\right)} + \left(\operatorname{im}{\left(y\right)}\right)^{2} - 6\right)^{2}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(- 2 \operatorname{re}{\left(y\right)} \operatorname{im}{\left(y\right)} - 6 \operatorname{im}{\left(y\right)},- \left(\operatorname{re}{\left(y\right)}\right)^{2} - 6 \operatorname{re}{\left(y\right)} + \left(\operatorname{im}{\left(y\right)}\right)^{2} - 6 \right)}}{2} \right)} - 3$$
x2 = i*((-2*re(y)*im(y) - 6*im(y))^2 + (-re(y)^2 - 6*re(y) + im(y)^2 - 6)^2)^(1/4)*sin(atan2(-2*re(y)*im(y) - 6*im(y, -re(y)^2 - 6*re(y) + im(y)^2 - 6)/2) + ((-2*re(y)*im(y) - 6*im(y))^2 + (-re(y)^2 - 6*re(y) + im(y)^2 - 6)^2)^(1/4)*cos(atan2(-2*re(y)*im(y) - 6*im(y), -re(y)^2 - 6*re(y) + im(y)^2 - 6)/2) - 3)
Sum and product of roots
[src]
sum
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/ 2 / / 2 2 \\ / 2 / / 2 2 \\ / 2 / / 2 2 \\ / 2 / / 2 2 \\
4 / 2 / 2 2 \ |atan2\-6*im(y) - 2*im(y)*re(y), -6 + im (y) - re (y) - 6*re(y)/| 4 / 2 / 2 2 \ |atan2\-6*im(y) - 2*im(y)*re(y), -6 + im (y) - re (y) - 6*re(y)/| 4 / 2 / 2 2 \ |atan2\-6*im(y) - 2*im(y)*re(y), -6 + im (y) - re (y) - 6*re(y)/| 4 / 2 / 2 2 \ |atan2\-6*im(y) - 2*im(y)*re(y), -6 + im (y) - re (y) - 6*re(y)/|
-3 - \/ (-6*im(y) - 2*im(y)*re(y)) + \-6 + im (y) - re (y) - 6*re(y)/ *cos|---------------------------------------------------------------| - I*\/ (-6*im(y) - 2*im(y)*re(y)) + \-6 + im (y) - re (y) - 6*re(y)/ *sin|---------------------------------------------------------------| + -3 + \/ (-6*im(y) - 2*im(y)*re(y)) + \-6 + im (y) - re (y) - 6*re(y)/ *cos|---------------------------------------------------------------| + I*\/ (-6*im(y) - 2*im(y)*re(y)) + \-6 + im (y) - re (y) - 6*re(y)/ *sin|---------------------------------------------------------------|
\ 2 / \ 2 / \ 2 / \ 2 /
$$\left(- i \sqrt[4]{\left(- 2 \operatorname{re}{\left(y\right)} \operatorname{im}{\left(y\right)} - 6 \operatorname{im}{\left(y\right)}\right)^{2} + \left(- \left(\operatorname{re}{\left(y\right)}\right)^{2} - 6 \operatorname{re}{\left(y\right)} + \left(\operatorname{im}{\left(y\right)}\right)^{2} - 6\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(- 2 \operatorname{re}{\left(y\right)} \operatorname{im}{\left(y\right)} - 6 \operatorname{im}{\left(y\right)},- \left(\operatorname{re}{\left(y\right)}\right)^{2} - 6 \operatorname{re}{\left(y\right)} + \left(\operatorname{im}{\left(y\right)}\right)^{2} - 6 \right)}}{2} \right)} - \sqrt[4]{\left(- 2 \operatorname{re}{\left(y\right)} \operatorname{im}{\left(y\right)} - 6 \operatorname{im}{\left(y\right)}\right)^{2} + \left(- \left(\operatorname{re}{\left(y\right)}\right)^{2} - 6 \operatorname{re}{\left(y\right)} + \left(\operatorname{im}{\left(y\right)}\right)^{2} - 6\right)^{2}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(- 2 \operatorname{re}{\left(y\right)} \operatorname{im}{\left(y\right)} - 6 \operatorname{im}{\left(y\right)},- \left(\operatorname{re}{\left(y\right)}\right)^{2} - 6 \operatorname{re}{\left(y\right)} + \left(\operatorname{im}{\left(y\right)}\right)^{2} - 6 \right)}}{2} \right)} - 3\right) + \left(i \sqrt[4]{\left(- 2 \operatorname{re}{\left(y\right)} \operatorname{im}{\left(y\right)} - 6 \operatorname{im}{\left(y\right)}\right)^{2} + \left(- \left(\operatorname{re}{\left(y\right)}\right)^{2} - 6 \operatorname{re}{\left(y\right)} + \left(\operatorname{im}{\left(y\right)}\right)^{2} - 6\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(- 2 \operatorname{re}{\left(y\right)} \operatorname{im}{\left(y\right)} - 6 \operatorname{im}{\left(y\right)},- \left(\operatorname{re}{\left(y\right)}\right)^{2} - 6 \operatorname{re}{\left(y\right)} + \left(\operatorname{im}{\left(y\right)}\right)^{2} - 6 \right)}}{2} \right)} + \sqrt[4]{\left(- 2 \operatorname{re}{\left(y\right)} \operatorname{im}{\left(y\right)} - 6 \operatorname{im}{\left(y\right)}\right)^{2} + \left(- \left(\operatorname{re}{\left(y\right)}\right)^{2} - 6 \operatorname{re}{\left(y\right)} + \left(\operatorname{im}{\left(y\right)}\right)^{2} - 6\right)^{2}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(- 2 \operatorname{re}{\left(y\right)} \operatorname{im}{\left(y\right)} - 6 \operatorname{im}{\left(y\right)},- \left(\operatorname{re}{\left(y\right)}\right)^{2} - 6 \operatorname{re}{\left(y\right)} + \left(\operatorname{im}{\left(y\right)}\right)^{2} - 6 \right)}}{2} \right)} - 3\right)$$
=
-6
$$-6$$
product
/ _________________________________________________________________ _________________________________________________________________ \ / _________________________________________________________________ _________________________________________________________________ \ | / 2 / / 2 2 \\ / 2 / / 2 2 \\| | / 2 / / 2 2 \\ / 2 / / 2 2 \\| | 4 / 2 / 2 2 \ |atan2\-6*im(y) - 2*im(y)*re(y), -6 + im (y) - re (y) - 6*re(y)/| 4 / 2 / 2 2 \ |atan2\-6*im(y) - 2*im(y)*re(y), -6 + im (y) - re (y) - 6*re(y)/|| | 4 / 2 / 2 2 \ |atan2\-6*im(y) - 2*im(y)*re(y), -6 + im (y) - re (y) - 6*re(y)/| 4 / 2 / 2 2 \ |atan2\-6*im(y) - 2*im(y)*re(y), -6 + im (y) - re (y) - 6*re(y)/|| |-3 - \/ (-6*im(y) - 2*im(y)*re(y)) + \-6 + im (y) - re (y) - 6*re(y)/ *cos|---------------------------------------------------------------| - I*\/ (-6*im(y) - 2*im(y)*re(y)) + \-6 + im (y) - re (y) - 6*re(y)/ *sin|---------------------------------------------------------------||*|-3 + \/ (-6*im(y) - 2*im(y)*re(y)) + \-6 + im (y) - re (y) - 6*re(y)/ *cos|---------------------------------------------------------------| + I*\/ (-6*im(y) - 2*im(y)*re(y)) + \-6 + im (y) - re (y) - 6*re(y)/ *sin|---------------------------------------------------------------|| \ \ 2 / \ 2 // \ \ 2 / \ 2 //
$$\left(- i \sqrt[4]{\left(- 2 \operatorname{re}{\left(y\right)} \operatorname{im}{\left(y\right)} - 6 \operatorname{im}{\left(y\right)}\right)^{2} + \left(- \left(\operatorname{re}{\left(y\right)}\right)^{2} - 6 \operatorname{re}{\left(y\right)} + \left(\operatorname{im}{\left(y\right)}\right)^{2} - 6\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(- 2 \operatorname{re}{\left(y\right)} \operatorname{im}{\left(y\right)} - 6 \operatorname{im}{\left(y\right)},- \left(\operatorname{re}{\left(y\right)}\right)^{2} - 6 \operatorname{re}{\left(y\right)} + \left(\operatorname{im}{\left(y\right)}\right)^{2} - 6 \right)}}{2} \right)} - \sqrt[4]{\left(- 2 \operatorname{re}{\left(y\right)} \operatorname{im}{\left(y\right)} - 6 \operatorname{im}{\left(y\right)}\right)^{2} + \left(- \left(\operatorname{re}{\left(y\right)}\right)^{2} - 6 \operatorname{re}{\left(y\right)} + \left(\operatorname{im}{\left(y\right)}\right)^{2} - 6\right)^{2}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(- 2 \operatorname{re}{\left(y\right)} \operatorname{im}{\left(y\right)} - 6 \operatorname{im}{\left(y\right)},- \left(\operatorname{re}{\left(y\right)}\right)^{2} - 6 \operatorname{re}{\left(y\right)} + \left(\operatorname{im}{\left(y\right)}\right)^{2} - 6 \right)}}{2} \right)} - 3\right) \left(i \sqrt[4]{\left(- 2 \operatorname{re}{\left(y\right)} \operatorname{im}{\left(y\right)} - 6 \operatorname{im}{\left(y\right)}\right)^{2} + \left(- \left(\operatorname{re}{\left(y\right)}\right)^{2} - 6 \operatorname{re}{\left(y\right)} + \left(\operatorname{im}{\left(y\right)}\right)^{2} - 6\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(- 2 \operatorname{re}{\left(y\right)} \operatorname{im}{\left(y\right)} - 6 \operatorname{im}{\left(y\right)},- \left(\operatorname{re}{\left(y\right)}\right)^{2} - 6 \operatorname{re}{\left(y\right)} + \left(\operatorname{im}{\left(y\right)}\right)^{2} - 6 \right)}}{2} \right)} + \sqrt[4]{\left(- 2 \operatorname{re}{\left(y\right)} \operatorname{im}{\left(y\right)} - 6 \operatorname{im}{\left(y\right)}\right)^{2} + \left(- \left(\operatorname{re}{\left(y\right)}\right)^{2} - 6 \operatorname{re}{\left(y\right)} + \left(\operatorname{im}{\left(y\right)}\right)^{2} - 6\right)^{2}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(- 2 \operatorname{re}{\left(y\right)} \operatorname{im}{\left(y\right)} - 6 \operatorname{im}{\left(y\right)},- \left(\operatorname{re}{\left(y\right)}\right)^{2} - 6 \operatorname{re}{\left(y\right)} + \left(\operatorname{im}{\left(y\right)}\right)^{2} - 6 \right)}}{2} \right)} - 3\right)$$
=
2 2 15 + re (y) - im (y) + 6*re(y) + 6*I*im(y) + 2*I*im(y)*re(y)
$$\left(\operatorname{re}{\left(y\right)}\right)^{2} + 2 i \operatorname{re}{\left(y\right)} \operatorname{im}{\left(y\right)} + 6 \operatorname{re}{\left(y\right)} - \left(\operatorname{im}{\left(y\right)}\right)^{2} + 6 i \operatorname{im}{\left(y\right)} + 15$$
15 + re(y)^2 - im(y)^2 + 6*re(y) + 6*i*im(y) + 2*i*im(y)*re(y)