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Similar expressions
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(x-1)^2+2(y-2)^2=1 equation
The teacher will be very surprised to see your correct solution 😉
The solution
You have entered
[src]
2 2 (x - 1) + 2*(y - 2) = 1
$$\left(x - 1\right)^{2} + 2 \left(y - 2\right)^{2} = 1$$
Detail solution
Move right part of the equation to
left part with negative sign.
The equation is transformed from
$$\left(x - 1\right)^{2} + 2 \left(y - 2\right)^{2} = 1$$
to
$$\left(\left(x - 1\right)^{2} + 2 \left(y - 2\right)^{2}\right) - 1 = 0$$
Expand the expression in the equation
$$\left(\left(x - 1\right)^{2} + 2 \left(y - 2\right)^{2}\right) - 1 = 0$$
We get the quadratic equation
$$x^{2} - 2 x + 2 y^{2} - 8 y + 8 = 0$$
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
$$a = 1$$
$$b = -2$$
$$c = 2 y^{2} - 8 y + 8$$
, then
The equation has two roots.
or
$$x_{1} = \frac{\sqrt{- 8 y^{2} + 32 y - 28}}{2} + 1$$
$$x_{2} = 1 - \frac{\sqrt{- 8 y^{2} + 32 y - 28}}{2}$$
left part with negative sign.
The equation is transformed from
$$\left(x - 1\right)^{2} + 2 \left(y - 2\right)^{2} = 1$$
to
$$\left(\left(x - 1\right)^{2} + 2 \left(y - 2\right)^{2}\right) - 1 = 0$$
Expand the expression in the equation
$$\left(\left(x - 1\right)^{2} + 2 \left(y - 2\right)^{2}\right) - 1 = 0$$
We get the quadratic equation
$$x^{2} - 2 x + 2 y^{2} - 8 y + 8 = 0$$
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
$$a = 1$$
$$b = -2$$
$$c = 2 y^{2} - 8 y + 8$$
, then
D = b^2 - 4 * a * c =
(-2)^2 - 4 * (1) * (8 - 8*y + 2*y^2) = -28 - 8*y^2 + 32*y
The equation has two roots.
x1 = (-b + sqrt(D)) / (2*a)
x2 = (-b - sqrt(D)) / (2*a)
or
$$x_{1} = \frac{\sqrt{- 8 y^{2} + 32 y - 28}}{2} + 1$$
$$x_{2} = 1 - \frac{\sqrt{- 8 y^{2} + 32 y - 28}}{2}$$
The graph
Rapid solution
[src]
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/ 2 / / 2 2 \\ / 2 / / 2 2 \\
4 / 2 / 2 2 \ |atan2\8*im(y) - 4*im(y)*re(y), -7 - 2*re (y) + 2*im (y) + 8*re(y)/| 4 / 2 / 2 2 \ |atan2\8*im(y) - 4*im(y)*re(y), -7 - 2*re (y) + 2*im (y) + 8*re(y)/|
x1 = 1 - \/ (8*im(y) - 4*im(y)*re(y)) + \-7 - 2*re (y) + 2*im (y) + 8*re(y)/ *cos|------------------------------------------------------------------| - I*\/ (8*im(y) - 4*im(y)*re(y)) + \-7 - 2*re (y) + 2*im (y) + 8*re(y)/ *sin|------------------------------------------------------------------|
\ 2 / \ 2 /
$$x_{1} = - i \sqrt[4]{\left(- 4 \operatorname{re}{\left(y\right)} \operatorname{im}{\left(y\right)} + 8 \operatorname{im}{\left(y\right)}\right)^{2} + \left(- 2 \left(\operatorname{re}{\left(y\right)}\right)^{2} + 8 \operatorname{re}{\left(y\right)} + 2 \left(\operatorname{im}{\left(y\right)}\right)^{2} - 7\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(- 4 \operatorname{re}{\left(y\right)} \operatorname{im}{\left(y\right)} + 8 \operatorname{im}{\left(y\right)},- 2 \left(\operatorname{re}{\left(y\right)}\right)^{2} + 8 \operatorname{re}{\left(y\right)} + 2 \left(\operatorname{im}{\left(y\right)}\right)^{2} - 7 \right)}}{2} \right)} - \sqrt[4]{\left(- 4 \operatorname{re}{\left(y\right)} \operatorname{im}{\left(y\right)} + 8 \operatorname{im}{\left(y\right)}\right)^{2} + \left(- 2 \left(\operatorname{re}{\left(y\right)}\right)^{2} + 8 \operatorname{re}{\left(y\right)} + 2 \left(\operatorname{im}{\left(y\right)}\right)^{2} - 7\right)^{2}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(- 4 \operatorname{re}{\left(y\right)} \operatorname{im}{\left(y\right)} + 8 \operatorname{im}{\left(y\right)},- 2 \left(\operatorname{re}{\left(y\right)}\right)^{2} + 8 \operatorname{re}{\left(y\right)} + 2 \left(\operatorname{im}{\left(y\right)}\right)^{2} - 7 \right)}}{2} \right)} + 1$$
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/ 2 / / 2 2 \\ / 2 / / 2 2 \\
4 / 2 / 2 2 \ |atan2\8*im(y) - 4*im(y)*re(y), -7 - 2*re (y) + 2*im (y) + 8*re(y)/| 4 / 2 / 2 2 \ |atan2\8*im(y) - 4*im(y)*re(y), -7 - 2*re (y) + 2*im (y) + 8*re(y)/|
x2 = 1 + \/ (8*im(y) - 4*im(y)*re(y)) + \-7 - 2*re (y) + 2*im (y) + 8*re(y)/ *cos|------------------------------------------------------------------| + I*\/ (8*im(y) - 4*im(y)*re(y)) + \-7 - 2*re (y) + 2*im (y) + 8*re(y)/ *sin|------------------------------------------------------------------|
\ 2 / \ 2 /
$$x_{2} = i \sqrt[4]{\left(- 4 \operatorname{re}{\left(y\right)} \operatorname{im}{\left(y\right)} + 8 \operatorname{im}{\left(y\right)}\right)^{2} + \left(- 2 \left(\operatorname{re}{\left(y\right)}\right)^{2} + 8 \operatorname{re}{\left(y\right)} + 2 \left(\operatorname{im}{\left(y\right)}\right)^{2} - 7\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(- 4 \operatorname{re}{\left(y\right)} \operatorname{im}{\left(y\right)} + 8 \operatorname{im}{\left(y\right)},- 2 \left(\operatorname{re}{\left(y\right)}\right)^{2} + 8 \operatorname{re}{\left(y\right)} + 2 \left(\operatorname{im}{\left(y\right)}\right)^{2} - 7 \right)}}{2} \right)} + \sqrt[4]{\left(- 4 \operatorname{re}{\left(y\right)} \operatorname{im}{\left(y\right)} + 8 \operatorname{im}{\left(y\right)}\right)^{2} + \left(- 2 \left(\operatorname{re}{\left(y\right)}\right)^{2} + 8 \operatorname{re}{\left(y\right)} + 2 \left(\operatorname{im}{\left(y\right)}\right)^{2} - 7\right)^{2}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(- 4 \operatorname{re}{\left(y\right)} \operatorname{im}{\left(y\right)} + 8 \operatorname{im}{\left(y\right)},- 2 \left(\operatorname{re}{\left(y\right)}\right)^{2} + 8 \operatorname{re}{\left(y\right)} + 2 \left(\operatorname{im}{\left(y\right)}\right)^{2} - 7 \right)}}{2} \right)} + 1$$
x2 = i*((-4*re(y)*im(y) + 8*im(y))^2 + (-2*re(y)^2 + 8*re(y) + 2*im(y)^2 - 7)^2)^(1/4)*sin(atan2(-4*re(y)*im(y) + 8*im(y, -2*re(y)^2 + 8*re(y) + 2*im(y)^2 - 7)/2) + ((-4*re(y)*im(y) + 8*im(y))^2 + (-2*re(y)^2 + 8*re(y) + 2*im(y)^2 - 7)^2)^(1/4)*cos(atan2(-4*re(y)*im(y) + 8*im(y), -2*re(y)^2 + 8*re(y) + 2*im(y)^2 - 7)/2) + 1)
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\8*im(y) - 4*im(y)*re(y), -7 - 2*re (y) + 2*im (y) + 8*re(y)/| 4 / 2 / 2 2 \ |atan2\8*im(y) - 4*im(y)*re(y), -7 - 2*re (y) + 2*im (y) + 8*re(y)/| 4 / 2 / 2 2 \ |atan2\8*im(y) - 4*im(y)*re(y), -7 - 2*re (y) + 2*im (y) + 8*re(y)/| 4 / 2 / 2 2 \ |atan2\8*im(y) - 4*im(y)*re(y), -7 - 2*re (y) + 2*im (y) + 8*re(y)/|
1 - \/ (8*im(y) - 4*im(y)*re(y)) + \-7 - 2*re (y) + 2*im (y) + 8*re(y)/ *cos|------------------------------------------------------------------| - I*\/ (8*im(y) - 4*im(y)*re(y)) + \-7 - 2*re (y) + 2*im (y) + 8*re(y)/ *sin|------------------------------------------------------------------| + 1 + \/ (8*im(y) - 4*im(y)*re(y)) + \-7 - 2*re (y) + 2*im (y) + 8*re(y)/ *cos|------------------------------------------------------------------| + I*\/ (8*im(y) - 4*im(y)*re(y)) + \-7 - 2*re (y) + 2*im (y) + 8*re(y)/ *sin|------------------------------------------------------------------|
\ 2 / \ 2 / \ 2 / \ 2 /
$$\left(- i \sqrt[4]{\left(- 4 \operatorname{re}{\left(y\right)} \operatorname{im}{\left(y\right)} + 8 \operatorname{im}{\left(y\right)}\right)^{2} + \left(- 2 \left(\operatorname{re}{\left(y\right)}\right)^{2} + 8 \operatorname{re}{\left(y\right)} + 2 \left(\operatorname{im}{\left(y\right)}\right)^{2} - 7\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(- 4 \operatorname{re}{\left(y\right)} \operatorname{im}{\left(y\right)} + 8 \operatorname{im}{\left(y\right)},- 2 \left(\operatorname{re}{\left(y\right)}\right)^{2} + 8 \operatorname{re}{\left(y\right)} + 2 \left(\operatorname{im}{\left(y\right)}\right)^{2} - 7 \right)}}{2} \right)} - \sqrt[4]{\left(- 4 \operatorname{re}{\left(y\right)} \operatorname{im}{\left(y\right)} + 8 \operatorname{im}{\left(y\right)}\right)^{2} + \left(- 2 \left(\operatorname{re}{\left(y\right)}\right)^{2} + 8 \operatorname{re}{\left(y\right)} + 2 \left(\operatorname{im}{\left(y\right)}\right)^{2} - 7\right)^{2}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(- 4 \operatorname{re}{\left(y\right)} \operatorname{im}{\left(y\right)} + 8 \operatorname{im}{\left(y\right)},- 2 \left(\operatorname{re}{\left(y\right)}\right)^{2} + 8 \operatorname{re}{\left(y\right)} + 2 \left(\operatorname{im}{\left(y\right)}\right)^{2} - 7 \right)}}{2} \right)} + 1\right) + \left(i \sqrt[4]{\left(- 4 \operatorname{re}{\left(y\right)} \operatorname{im}{\left(y\right)} + 8 \operatorname{im}{\left(y\right)}\right)^{2} + \left(- 2 \left(\operatorname{re}{\left(y\right)}\right)^{2} + 8 \operatorname{re}{\left(y\right)} + 2 \left(\operatorname{im}{\left(y\right)}\right)^{2} - 7\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(- 4 \operatorname{re}{\left(y\right)} \operatorname{im}{\left(y\right)} + 8 \operatorname{im}{\left(y\right)},- 2 \left(\operatorname{re}{\left(y\right)}\right)^{2} + 8 \operatorname{re}{\left(y\right)} + 2 \left(\operatorname{im}{\left(y\right)}\right)^{2} - 7 \right)}}{2} \right)} + \sqrt[4]{\left(- 4 \operatorname{re}{\left(y\right)} \operatorname{im}{\left(y\right)} + 8 \operatorname{im}{\left(y\right)}\right)^{2} + \left(- 2 \left(\operatorname{re}{\left(y\right)}\right)^{2} + 8 \operatorname{re}{\left(y\right)} + 2 \left(\operatorname{im}{\left(y\right)}\right)^{2} - 7\right)^{2}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(- 4 \operatorname{re}{\left(y\right)} \operatorname{im}{\left(y\right)} + 8 \operatorname{im}{\left(y\right)},- 2 \left(\operatorname{re}{\left(y\right)}\right)^{2} + 8 \operatorname{re}{\left(y\right)} + 2 \left(\operatorname{im}{\left(y\right)}\right)^{2} - 7 \right)}}{2} \right)} + 1\right)$$
=
2
$$2$$
product
/ ____________________________________________________________________ ____________________________________________________________________ \ / ____________________________________________________________________ ____________________________________________________________________ \ | / 2 / / 2 2 \\ / 2 / / 2 2 \\| | / 2 / / 2 2 \\ / 2 / / 2 2 \\| | 4 / 2 / 2 2 \ |atan2\8*im(y) - 4*im(y)*re(y), -7 - 2*re (y) + 2*im (y) + 8*re(y)/| 4 / 2 / 2 2 \ |atan2\8*im(y) - 4*im(y)*re(y), -7 - 2*re (y) + 2*im (y) + 8*re(y)/|| | 4 / 2 / 2 2 \ |atan2\8*im(y) - 4*im(y)*re(y), -7 - 2*re (y) + 2*im (y) + 8*re(y)/| 4 / 2 / 2 2 \ |atan2\8*im(y) - 4*im(y)*re(y), -7 - 2*re (y) + 2*im (y) + 8*re(y)/|| |1 - \/ (8*im(y) - 4*im(y)*re(y)) + \-7 - 2*re (y) + 2*im (y) + 8*re(y)/ *cos|------------------------------------------------------------------| - I*\/ (8*im(y) - 4*im(y)*re(y)) + \-7 - 2*re (y) + 2*im (y) + 8*re(y)/ *sin|------------------------------------------------------------------||*|1 + \/ (8*im(y) - 4*im(y)*re(y)) + \-7 - 2*re (y) + 2*im (y) + 8*re(y)/ *cos|------------------------------------------------------------------| + I*\/ (8*im(y) - 4*im(y)*re(y)) + \-7 - 2*re (y) + 2*im (y) + 8*re(y)/ *sin|------------------------------------------------------------------|| \ \ 2 / \ 2 // \ \ 2 / \ 2 //
$$\left(- i \sqrt[4]{\left(- 4 \operatorname{re}{\left(y\right)} \operatorname{im}{\left(y\right)} + 8 \operatorname{im}{\left(y\right)}\right)^{2} + \left(- 2 \left(\operatorname{re}{\left(y\right)}\right)^{2} + 8 \operatorname{re}{\left(y\right)} + 2 \left(\operatorname{im}{\left(y\right)}\right)^{2} - 7\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(- 4 \operatorname{re}{\left(y\right)} \operatorname{im}{\left(y\right)} + 8 \operatorname{im}{\left(y\right)},- 2 \left(\operatorname{re}{\left(y\right)}\right)^{2} + 8 \operatorname{re}{\left(y\right)} + 2 \left(\operatorname{im}{\left(y\right)}\right)^{2} - 7 \right)}}{2} \right)} - \sqrt[4]{\left(- 4 \operatorname{re}{\left(y\right)} \operatorname{im}{\left(y\right)} + 8 \operatorname{im}{\left(y\right)}\right)^{2} + \left(- 2 \left(\operatorname{re}{\left(y\right)}\right)^{2} + 8 \operatorname{re}{\left(y\right)} + 2 \left(\operatorname{im}{\left(y\right)}\right)^{2} - 7\right)^{2}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(- 4 \operatorname{re}{\left(y\right)} \operatorname{im}{\left(y\right)} + 8 \operatorname{im}{\left(y\right)},- 2 \left(\operatorname{re}{\left(y\right)}\right)^{2} + 8 \operatorname{re}{\left(y\right)} + 2 \left(\operatorname{im}{\left(y\right)}\right)^{2} - 7 \right)}}{2} \right)} + 1\right) \left(i \sqrt[4]{\left(- 4 \operatorname{re}{\left(y\right)} \operatorname{im}{\left(y\right)} + 8 \operatorname{im}{\left(y\right)}\right)^{2} + \left(- 2 \left(\operatorname{re}{\left(y\right)}\right)^{2} + 8 \operatorname{re}{\left(y\right)} + 2 \left(\operatorname{im}{\left(y\right)}\right)^{2} - 7\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(- 4 \operatorname{re}{\left(y\right)} \operatorname{im}{\left(y\right)} + 8 \operatorname{im}{\left(y\right)},- 2 \left(\operatorname{re}{\left(y\right)}\right)^{2} + 8 \operatorname{re}{\left(y\right)} + 2 \left(\operatorname{im}{\left(y\right)}\right)^{2} - 7 \right)}}{2} \right)} + \sqrt[4]{\left(- 4 \operatorname{re}{\left(y\right)} \operatorname{im}{\left(y\right)} + 8 \operatorname{im}{\left(y\right)}\right)^{2} + \left(- 2 \left(\operatorname{re}{\left(y\right)}\right)^{2} + 8 \operatorname{re}{\left(y\right)} + 2 \left(\operatorname{im}{\left(y\right)}\right)^{2} - 7\right)^{2}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(- 4 \operatorname{re}{\left(y\right)} \operatorname{im}{\left(y\right)} + 8 \operatorname{im}{\left(y\right)},- 2 \left(\operatorname{re}{\left(y\right)}\right)^{2} + 8 \operatorname{re}{\left(y\right)} + 2 \left(\operatorname{im}{\left(y\right)}\right)^{2} - 7 \right)}}{2} \right)} + 1\right)$$
=
2 2 8 - 8*re(y) - 2*im (y) + 2*re (y) - 8*I*im(y) + 4*I*im(y)*re(y)
$$2 \left(\operatorname{re}{\left(y\right)}\right)^{2} + 4 i \operatorname{re}{\left(y\right)} \operatorname{im}{\left(y\right)} - 8 \operatorname{re}{\left(y\right)} - 2 \left(\operatorname{im}{\left(y\right)}\right)^{2} - 8 i \operatorname{im}{\left(y\right)} + 8$$
8 - 8*re(y) - 2*im(y)^2 + 2*re(y)^2 - 8*i*im(y) + 4*i*im(y)*re(y)