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(1/(x-4)^2)-(7/(x-4))+10=y equation
The teacher will be very surprised to see your correct solution 😉
The solution
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[src]
1 7
-------- - ----- + 10 = y
2 x - 4
(x - 4)
$$\left(\frac{1}{\left(x - 4\right)^{2}} - \frac{7}{x - 4}\right) + 10 = y$$
Detail solution
Given the equation:
$$\left(\frac{1}{\left(x - 4\right)^{2}} - \frac{7}{x - 4}\right) + 10 = y$$
Multiply the equation sides by the denominators:
(-4 + x)^2
we get:
$$\left(x - 4\right)^{2} \left(\left(\frac{1}{\left(x - 4\right)^{2}} - \frac{7}{x - 4}\right) + 10\right) = y \left(x - 4\right)^{2}$$
$$10 x^{2} - 87 x + 189 = y \left(x - 4\right)^{2}$$
Move right part of the equation to
left part with negative sign.
The equation is transformed from
$$10 x^{2} - 87 x + 189 = y \left(x - 4\right)^{2}$$
to
$$- x^{2} y + 10 x^{2} + 8 x y - 87 x - 16 y + 189 = 0$$
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
$$a = 10 - y$$
$$b = 8 y - 87$$
$$c = 189 - 16 y$$
, then
The equation has two roots.
or
$$x_{1} = \frac{- 8 y + \sqrt{- \left(40 - 4 y\right) \left(189 - 16 y\right) + \left(8 y - 87\right)^{2}} + 87}{20 - 2 y}$$
$$x_{2} = \frac{- 8 y - \sqrt{- \left(40 - 4 y\right) \left(189 - 16 y\right) + \left(8 y - 87\right)^{2}} + 87}{20 - 2 y}$$
$$\left(\frac{1}{\left(x - 4\right)^{2}} - \frac{7}{x - 4}\right) + 10 = y$$
Multiply the equation sides by the denominators:
(-4 + x)^2
we get:
$$\left(x - 4\right)^{2} \left(\left(\frac{1}{\left(x - 4\right)^{2}} - \frac{7}{x - 4}\right) + 10\right) = y \left(x - 4\right)^{2}$$
$$10 x^{2} - 87 x + 189 = y \left(x - 4\right)^{2}$$
Move right part of the equation to
left part with negative sign.
The equation is transformed from
$$10 x^{2} - 87 x + 189 = y \left(x - 4\right)^{2}$$
to
$$- x^{2} y + 10 x^{2} + 8 x y - 87 x - 16 y + 189 = 0$$
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
$$a = 10 - y$$
$$b = 8 y - 87$$
$$c = 189 - 16 y$$
, then
D = b^2 - 4 * a * c =
(-87 + 8*y)^2 - 4 * (10 - y) * (189 - 16*y) = (-87 + 8*y)^2 - (40 - 4*y)*(189 - 16*y)
The equation has two roots.
x1 = (-b + sqrt(D)) / (2*a)
x2 = (-b - sqrt(D)) / (2*a)
or
$$x_{1} = \frac{- 8 y + \sqrt{- \left(40 - 4 y\right) \left(189 - 16 y\right) + \left(8 y - 87\right)^{2}} + 87}{20 - 2 y}$$
$$x_{2} = \frac{- 8 y - \sqrt{- \left(40 - 4 y\right) \left(189 - 16 y\right) + \left(8 y - 87\right)^{2}} + 87}{20 - 2 y}$$
The graph
Rapid solution
[src]
/ / ____________________________ \ / ____________________________ \ \ / ____________________________ \ / ____________________________ \
| | 4 / 2 2 /atan2(4*im(y), 9 + 4*re(y))\| | 4 / 2 2 /atan2(4*im(y), 9 + 4*re(y))\| | | 4 / 2 2 /atan2(4*im(y), 9 + 4*re(y))\| | 4 / 2 2 /atan2(4*im(y), 9 + 4*re(y))\|
|(-20 + 2*re(y))*|8*im(y) - \/ (9 + 4*re(y)) + 16*im (y) *sin|---------------------------|| 2*|-87 + 8*re(y) - \/ (9 + 4*re(y)) + 16*im (y) *cos|---------------------------||*im(y)| (-20 + 2*re(y))*|-87 + 8*re(y) - \/ (9 + 4*re(y)) + 16*im (y) *cos|---------------------------|| 2*|8*im(y) - \/ (9 + 4*re(y)) + 16*im (y) *sin|---------------------------||*im(y)
| \ \ 2 // \ \ 2 // | \ \ 2 // \ \ 2 //
x1 = I*|-------------------------------------------------------------------------------------------- - ------------------------------------------------------------------------------------------| + -------------------------------------------------------------------------------------------------- + ------------------------------------------------------------------------------------
| 2 2 2 2 | 2 2 2 2
\ (-20 + 2*re(y)) + 4*im (y) (-20 + 2*re(y)) + 4*im (y) / (-20 + 2*re(y)) + 4*im (y) (-20 + 2*re(y)) + 4*im (y)
$$x_{1} = \frac{2 \left(- \sqrt[4]{\left(4 \operatorname{re}{\left(y\right)} + 9\right)^{2} + 16 \left(\operatorname{im}{\left(y\right)}\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(4 \operatorname{im}{\left(y\right)},4 \operatorname{re}{\left(y\right)} + 9 \right)}}{2} \right)} + 8 \operatorname{im}{\left(y\right)}\right) \operatorname{im}{\left(y\right)}}{\left(2 \operatorname{re}{\left(y\right)} - 20\right)^{2} + 4 \left(\operatorname{im}{\left(y\right)}\right)^{2}} + i \left(\frac{\left(- \sqrt[4]{\left(4 \operatorname{re}{\left(y\right)} + 9\right)^{2} + 16 \left(\operatorname{im}{\left(y\right)}\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(4 \operatorname{im}{\left(y\right)},4 \operatorname{re}{\left(y\right)} + 9 \right)}}{2} \right)} + 8 \operatorname{im}{\left(y\right)}\right) \left(2 \operatorname{re}{\left(y\right)} - 20\right)}{\left(2 \operatorname{re}{\left(y\right)} - 20\right)^{2} + 4 \left(\operatorname{im}{\left(y\right)}\right)^{2}} - \frac{2 \left(- \sqrt[4]{\left(4 \operatorname{re}{\left(y\right)} + 9\right)^{2} + 16 \left(\operatorname{im}{\left(y\right)}\right)^{2}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(4 \operatorname{im}{\left(y\right)},4 \operatorname{re}{\left(y\right)} + 9 \right)}}{2} \right)} + 8 \operatorname{re}{\left(y\right)} - 87\right) \operatorname{im}{\left(y\right)}}{\left(2 \operatorname{re}{\left(y\right)} - 20\right)^{2} + 4 \left(\operatorname{im}{\left(y\right)}\right)^{2}}\right) + \frac{\left(2 \operatorname{re}{\left(y\right)} - 20\right) \left(- \sqrt[4]{\left(4 \operatorname{re}{\left(y\right)} + 9\right)^{2} + 16 \left(\operatorname{im}{\left(y\right)}\right)^{2}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(4 \operatorname{im}{\left(y\right)},4 \operatorname{re}{\left(y\right)} + 9 \right)}}{2} \right)} + 8 \operatorname{re}{\left(y\right)} - 87\right)}{\left(2 \operatorname{re}{\left(y\right)} - 20\right)^{2} + 4 \left(\operatorname{im}{\left(y\right)}\right)^{2}}$$
/ / ____________________________ \ / ____________________________ \ \ / ____________________________ \ / ____________________________ \
| | 4 / 2 2 /atan2(4*im(y), 9 + 4*re(y))\| | 4 / 2 2 /atan2(4*im(y), 9 + 4*re(y))\| | | 4 / 2 2 /atan2(4*im(y), 9 + 4*re(y))\| | 4 / 2 2 /atan2(4*im(y), 9 + 4*re(y))\|
|(-20 + 2*re(y))*|8*im(y) + \/ (9 + 4*re(y)) + 16*im (y) *sin|---------------------------|| 2*|-87 + 8*re(y) + \/ (9 + 4*re(y)) + 16*im (y) *cos|---------------------------||*im(y)| (-20 + 2*re(y))*|-87 + 8*re(y) + \/ (9 + 4*re(y)) + 16*im (y) *cos|---------------------------|| 2*|8*im(y) + \/ (9 + 4*re(y)) + 16*im (y) *sin|---------------------------||*im(y)
| \ \ 2 // \ \ 2 // | \ \ 2 // \ \ 2 //
x2 = I*|-------------------------------------------------------------------------------------------- - ------------------------------------------------------------------------------------------| + -------------------------------------------------------------------------------------------------- + ------------------------------------------------------------------------------------
| 2 2 2 2 | 2 2 2 2
\ (-20 + 2*re(y)) + 4*im (y) (-20 + 2*re(y)) + 4*im (y) / (-20 + 2*re(y)) + 4*im (y) (-20 + 2*re(y)) + 4*im (y)
$$x_{2} = \frac{2 \left(\sqrt[4]{\left(4 \operatorname{re}{\left(y\right)} + 9\right)^{2} + 16 \left(\operatorname{im}{\left(y\right)}\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(4 \operatorname{im}{\left(y\right)},4 \operatorname{re}{\left(y\right)} + 9 \right)}}{2} \right)} + 8 \operatorname{im}{\left(y\right)}\right) \operatorname{im}{\left(y\right)}}{\left(2 \operatorname{re}{\left(y\right)} - 20\right)^{2} + 4 \left(\operatorname{im}{\left(y\right)}\right)^{2}} + i \left(\frac{\left(\sqrt[4]{\left(4 \operatorname{re}{\left(y\right)} + 9\right)^{2} + 16 \left(\operatorname{im}{\left(y\right)}\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(4 \operatorname{im}{\left(y\right)},4 \operatorname{re}{\left(y\right)} + 9 \right)}}{2} \right)} + 8 \operatorname{im}{\left(y\right)}\right) \left(2 \operatorname{re}{\left(y\right)} - 20\right)}{\left(2 \operatorname{re}{\left(y\right)} - 20\right)^{2} + 4 \left(\operatorname{im}{\left(y\right)}\right)^{2}} - \frac{2 \left(\sqrt[4]{\left(4 \operatorname{re}{\left(y\right)} + 9\right)^{2} + 16 \left(\operatorname{im}{\left(y\right)}\right)^{2}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(4 \operatorname{im}{\left(y\right)},4 \operatorname{re}{\left(y\right)} + 9 \right)}}{2} \right)} + 8 \operatorname{re}{\left(y\right)} - 87\right) \operatorname{im}{\left(y\right)}}{\left(2 \operatorname{re}{\left(y\right)} - 20\right)^{2} + 4 \left(\operatorname{im}{\left(y\right)}\right)^{2}}\right) + \frac{\left(2 \operatorname{re}{\left(y\right)} - 20\right) \left(\sqrt[4]{\left(4 \operatorname{re}{\left(y\right)} + 9\right)^{2} + 16 \left(\operatorname{im}{\left(y\right)}\right)^{2}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(4 \operatorname{im}{\left(y\right)},4 \operatorname{re}{\left(y\right)} + 9 \right)}}{2} \right)} + 8 \operatorname{re}{\left(y\right)} - 87\right)}{\left(2 \operatorname{re}{\left(y\right)} - 20\right)^{2} + 4 \left(\operatorname{im}{\left(y\right)}\right)^{2}}$$
x2 = 2*(((4*re(y) + 9)^2 + 16*im(y)^2)^(1/4)*sin(atan2(4*im(y, 4*re(y) + 9)/2) + 8*im(y))*im(y)/((2*re(y) - 20)^2 + 4*im(y)^2) + i*((((4*re(y) + 9)^2 + 16*im(y)^2)^(1/4)*sin(atan2(4*im(y), 4*re(y) + 9)/2) + 8*im(y))*(2*re(y) - 20)/((2*re(y) - 20)^2 + 4*im(y)^2) - 2*(((4*re(y) + 9)^2 + 16*im(y)^2)^(1/4)*cos(atan2(4*im(y), 4*re(y) + 9)/2) + 8*re(y) - 87)*im(y)/((2*re(y) - 20)^2 + 4*im(y)^2)) + (2*re(y) - 20)*(((4*re(y) + 9)^2 + 16*im(y)^2)^(1/4)*cos(atan2(4*im(y), 4*re(y) + 9)/2) + 8*re(y) - 87)/((2*re(y) - 20)^2 + 4*im(y)^2))
Sum and product of roots
[src]
sum
/ / ____________________________ \ / ____________________________ \ \ / ____________________________ \ / ____________________________ \ / / ____________________________ \ / ____________________________ \ \ / ____________________________ \ / ____________________________ \ | | 4 / 2 2 /atan2(4*im(y), 9 + 4*re(y))\| | 4 / 2 2 /atan2(4*im(y), 9 + 4*re(y))\| | | 4 / 2 2 /atan2(4*im(y), 9 + 4*re(y))\| | 4 / 2 2 /atan2(4*im(y), 9 + 4*re(y))\| | | 4 / 2 2 /atan2(4*im(y), 9 + 4*re(y))\| | 4 / 2 2 /atan2(4*im(y), 9 + 4*re(y))\| | | 4 / 2 2 /atan2(4*im(y), 9 + 4*re(y))\| | 4 / 2 2 /atan2(4*im(y), 9 + 4*re(y))\| |(-20 + 2*re(y))*|8*im(y) - \/ (9 + 4*re(y)) + 16*im (y) *sin|---------------------------|| 2*|-87 + 8*re(y) - \/ (9 + 4*re(y)) + 16*im (y) *cos|---------------------------||*im(y)| (-20 + 2*re(y))*|-87 + 8*re(y) - \/ (9 + 4*re(y)) + 16*im (y) *cos|---------------------------|| 2*|8*im(y) - \/ (9 + 4*re(y)) + 16*im (y) *sin|---------------------------||*im(y) |(-20 + 2*re(y))*|8*im(y) + \/ (9 + 4*re(y)) + 16*im (y) *sin|---------------------------|| 2*|-87 + 8*re(y) + \/ (9 + 4*re(y)) + 16*im (y) *cos|---------------------------||*im(y)| (-20 + 2*re(y))*|-87 + 8*re(y) + \/ (9 + 4*re(y)) + 16*im (y) *cos|---------------------------|| 2*|8*im(y) + \/ (9 + 4*re(y)) + 16*im (y) *sin|---------------------------||*im(y) | \ \ 2 // \ \ 2 // | \ \ 2 // \ \ 2 // | \ \ 2 // \ \ 2 // | \ \ 2 // \ \ 2 // I*|-------------------------------------------------------------------------------------------- - ------------------------------------------------------------------------------------------| + -------------------------------------------------------------------------------------------------- + ------------------------------------------------------------------------------------ + I*|-------------------------------------------------------------------------------------------- - ------------------------------------------------------------------------------------------| + -------------------------------------------------------------------------------------------------- + ------------------------------------------------------------------------------------ | 2 2 2 2 | 2 2 2 2 | 2 2 2 2 | 2 2 2 2 \ (-20 + 2*re(y)) + 4*im (y) (-20 + 2*re(y)) + 4*im (y) / (-20 + 2*re(y)) + 4*im (y) (-20 + 2*re(y)) + 4*im (y) \ (-20 + 2*re(y)) + 4*im (y) (-20 + 2*re(y)) + 4*im (y) / (-20 + 2*re(y)) + 4*im (y) (-20 + 2*re(y)) + 4*im (y)
$$\left(\frac{2 \left(- \sqrt[4]{\left(4 \operatorname{re}{\left(y\right)} + 9\right)^{2} + 16 \left(\operatorname{im}{\left(y\right)}\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(4 \operatorname{im}{\left(y\right)},4 \operatorname{re}{\left(y\right)} + 9 \right)}}{2} \right)} + 8 \operatorname{im}{\left(y\right)}\right) \operatorname{im}{\left(y\right)}}{\left(2 \operatorname{re}{\left(y\right)} - 20\right)^{2} + 4 \left(\operatorname{im}{\left(y\right)}\right)^{2}} + i \left(\frac{\left(- \sqrt[4]{\left(4 \operatorname{re}{\left(y\right)} + 9\right)^{2} + 16 \left(\operatorname{im}{\left(y\right)}\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(4 \operatorname{im}{\left(y\right)},4 \operatorname{re}{\left(y\right)} + 9 \right)}}{2} \right)} + 8 \operatorname{im}{\left(y\right)}\right) \left(2 \operatorname{re}{\left(y\right)} - 20\right)}{\left(2 \operatorname{re}{\left(y\right)} - 20\right)^{2} + 4 \left(\operatorname{im}{\left(y\right)}\right)^{2}} - \frac{2 \left(- \sqrt[4]{\left(4 \operatorname{re}{\left(y\right)} + 9\right)^{2} + 16 \left(\operatorname{im}{\left(y\right)}\right)^{2}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(4 \operatorname{im}{\left(y\right)},4 \operatorname{re}{\left(y\right)} + 9 \right)}}{2} \right)} + 8 \operatorname{re}{\left(y\right)} - 87\right) \operatorname{im}{\left(y\right)}}{\left(2 \operatorname{re}{\left(y\right)} - 20\right)^{2} + 4 \left(\operatorname{im}{\left(y\right)}\right)^{2}}\right) + \frac{\left(2 \operatorname{re}{\left(y\right)} - 20\right) \left(- \sqrt[4]{\left(4 \operatorname{re}{\left(y\right)} + 9\right)^{2} + 16 \left(\operatorname{im}{\left(y\right)}\right)^{2}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(4 \operatorname{im}{\left(y\right)},4 \operatorname{re}{\left(y\right)} + 9 \right)}}{2} \right)} + 8 \operatorname{re}{\left(y\right)} - 87\right)}{\left(2 \operatorname{re}{\left(y\right)} - 20\right)^{2} + 4 \left(\operatorname{im}{\left(y\right)}\right)^{2}}\right) + \left(\frac{2 \left(\sqrt[4]{\left(4 \operatorname{re}{\left(y\right)} + 9\right)^{2} + 16 \left(\operatorname{im}{\left(y\right)}\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(4 \operatorname{im}{\left(y\right)},4 \operatorname{re}{\left(y\right)} + 9 \right)}}{2} \right)} + 8 \operatorname{im}{\left(y\right)}\right) \operatorname{im}{\left(y\right)}}{\left(2 \operatorname{re}{\left(y\right)} - 20\right)^{2} + 4 \left(\operatorname{im}{\left(y\right)}\right)^{2}} + i \left(\frac{\left(\sqrt[4]{\left(4 \operatorname{re}{\left(y\right)} + 9\right)^{2} + 16 \left(\operatorname{im}{\left(y\right)}\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(4 \operatorname{im}{\left(y\right)},4 \operatorname{re}{\left(y\right)} + 9 \right)}}{2} \right)} + 8 \operatorname{im}{\left(y\right)}\right) \left(2 \operatorname{re}{\left(y\right)} - 20\right)}{\left(2 \operatorname{re}{\left(y\right)} - 20\right)^{2} + 4 \left(\operatorname{im}{\left(y\right)}\right)^{2}} - \frac{2 \left(\sqrt[4]{\left(4 \operatorname{re}{\left(y\right)} + 9\right)^{2} + 16 \left(\operatorname{im}{\left(y\right)}\right)^{2}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(4 \operatorname{im}{\left(y\right)},4 \operatorname{re}{\left(y\right)} + 9 \right)}}{2} \right)} + 8 \operatorname{re}{\left(y\right)} - 87\right) \operatorname{im}{\left(y\right)}}{\left(2 \operatorname{re}{\left(y\right)} - 20\right)^{2} + 4 \left(\operatorname{im}{\left(y\right)}\right)^{2}}\right) + \frac{\left(2 \operatorname{re}{\left(y\right)} - 20\right) \left(\sqrt[4]{\left(4 \operatorname{re}{\left(y\right)} + 9\right)^{2} + 16 \left(\operatorname{im}{\left(y\right)}\right)^{2}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(4 \operatorname{im}{\left(y\right)},4 \operatorname{re}{\left(y\right)} + 9 \right)}}{2} \right)} + 8 \operatorname{re}{\left(y\right)} - 87\right)}{\left(2 \operatorname{re}{\left(y\right)} - 20\right)^{2} + 4 \left(\operatorname{im}{\left(y\right)}\right)^{2}}\right)$$
=
/ / ____________________________ \ / ____________________________ \ \ / / ____________________________ \ / ____________________________ \ \ / ____________________________ \ / ____________________________ \ / ____________________________ \ / ____________________________ \ | | 4 / 2 2 /atan2(4*im(y), 9 + 4*re(y))\| | 4 / 2 2 /atan2(4*im(y), 9 + 4*re(y))\| | | | 4 / 2 2 /atan2(4*im(y), 9 + 4*re(y))\| | 4 / 2 2 /atan2(4*im(y), 9 + 4*re(y))\| | | 4 / 2 2 /atan2(4*im(y), 9 + 4*re(y))\| | 4 / 2 2 /atan2(4*im(y), 9 + 4*re(y))\| | 4 / 2 2 /atan2(4*im(y), 9 + 4*re(y))\| | 4 / 2 2 /atan2(4*im(y), 9 + 4*re(y))\| |(-20 + 2*re(y))*|8*im(y) + \/ (9 + 4*re(y)) + 16*im (y) *sin|---------------------------|| 2*|-87 + 8*re(y) + \/ (9 + 4*re(y)) + 16*im (y) *cos|---------------------------||*im(y)| |(-20 + 2*re(y))*|8*im(y) - \/ (9 + 4*re(y)) + 16*im (y) *sin|---------------------------|| 2*|-87 + 8*re(y) - \/ (9 + 4*re(y)) + 16*im (y) *cos|---------------------------||*im(y)| (-20 + 2*re(y))*|-87 + 8*re(y) + \/ (9 + 4*re(y)) + 16*im (y) *cos|---------------------------|| (-20 + 2*re(y))*|-87 + 8*re(y) - \/ (9 + 4*re(y)) + 16*im (y) *cos|---------------------------|| 2*|8*im(y) + \/ (9 + 4*re(y)) + 16*im (y) *sin|---------------------------||*im(y) 2*|8*im(y) - \/ (9 + 4*re(y)) + 16*im (y) *sin|---------------------------||*im(y) | \ \ 2 // \ \ 2 // | | \ \ 2 // \ \ 2 // | \ \ 2 // \ \ 2 // \ \ 2 // \ \ 2 // I*|-------------------------------------------------------------------------------------------- - ------------------------------------------------------------------------------------------| + I*|-------------------------------------------------------------------------------------------- - ------------------------------------------------------------------------------------------| + -------------------------------------------------------------------------------------------------- + -------------------------------------------------------------------------------------------------- + ------------------------------------------------------------------------------------ + ------------------------------------------------------------------------------------ | 2 2 2 2 | | 2 2 2 2 | 2 2 2 2 2 2 2 2 \ (-20 + 2*re(y)) + 4*im (y) (-20 + 2*re(y)) + 4*im (y) / \ (-20 + 2*re(y)) + 4*im (y) (-20 + 2*re(y)) + 4*im (y) / (-20 + 2*re(y)) + 4*im (y) (-20 + 2*re(y)) + 4*im (y) (-20 + 2*re(y)) + 4*im (y) (-20 + 2*re(y)) + 4*im (y)
$$\frac{2 \left(- \sqrt[4]{\left(4 \operatorname{re}{\left(y\right)} + 9\right)^{2} + 16 \left(\operatorname{im}{\left(y\right)}\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(4 \operatorname{im}{\left(y\right)},4 \operatorname{re}{\left(y\right)} + 9 \right)}}{2} \right)} + 8 \operatorname{im}{\left(y\right)}\right) \operatorname{im}{\left(y\right)}}{\left(2 \operatorname{re}{\left(y\right)} - 20\right)^{2} + 4 \left(\operatorname{im}{\left(y\right)}\right)^{2}} + \frac{2 \left(\sqrt[4]{\left(4 \operatorname{re}{\left(y\right)} + 9\right)^{2} + 16 \left(\operatorname{im}{\left(y\right)}\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(4 \operatorname{im}{\left(y\right)},4 \operatorname{re}{\left(y\right)} + 9 \right)}}{2} \right)} + 8 \operatorname{im}{\left(y\right)}\right) \operatorname{im}{\left(y\right)}}{\left(2 \operatorname{re}{\left(y\right)} - 20\right)^{2} + 4 \left(\operatorname{im}{\left(y\right)}\right)^{2}} + i \left(\frac{\left(- \sqrt[4]{\left(4 \operatorname{re}{\left(y\right)} + 9\right)^{2} + 16 \left(\operatorname{im}{\left(y\right)}\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(4 \operatorname{im}{\left(y\right)},4 \operatorname{re}{\left(y\right)} + 9 \right)}}{2} \right)} + 8 \operatorname{im}{\left(y\right)}\right) \left(2 \operatorname{re}{\left(y\right)} - 20\right)}{\left(2 \operatorname{re}{\left(y\right)} - 20\right)^{2} + 4 \left(\operatorname{im}{\left(y\right)}\right)^{2}} - \frac{2 \left(- \sqrt[4]{\left(4 \operatorname{re}{\left(y\right)} + 9\right)^{2} + 16 \left(\operatorname{im}{\left(y\right)}\right)^{2}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(4 \operatorname{im}{\left(y\right)},4 \operatorname{re}{\left(y\right)} + 9 \right)}}{2} \right)} + 8 \operatorname{re}{\left(y\right)} - 87\right) \operatorname{im}{\left(y\right)}}{\left(2 \operatorname{re}{\left(y\right)} - 20\right)^{2} + 4 \left(\operatorname{im}{\left(y\right)}\right)^{2}}\right) + i \left(\frac{\left(\sqrt[4]{\left(4 \operatorname{re}{\left(y\right)} + 9\right)^{2} + 16 \left(\operatorname{im}{\left(y\right)}\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(4 \operatorname{im}{\left(y\right)},4 \operatorname{re}{\left(y\right)} + 9 \right)}}{2} \right)} + 8 \operatorname{im}{\left(y\right)}\right) \left(2 \operatorname{re}{\left(y\right)} - 20\right)}{\left(2 \operatorname{re}{\left(y\right)} - 20\right)^{2} + 4 \left(\operatorname{im}{\left(y\right)}\right)^{2}} - \frac{2 \left(\sqrt[4]{\left(4 \operatorname{re}{\left(y\right)} + 9\right)^{2} + 16 \left(\operatorname{im}{\left(y\right)}\right)^{2}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(4 \operatorname{im}{\left(y\right)},4 \operatorname{re}{\left(y\right)} + 9 \right)}}{2} \right)} + 8 \operatorname{re}{\left(y\right)} - 87\right) \operatorname{im}{\left(y\right)}}{\left(2 \operatorname{re}{\left(y\right)} - 20\right)^{2} + 4 \left(\operatorname{im}{\left(y\right)}\right)^{2}}\right) + \frac{\left(2 \operatorname{re}{\left(y\right)} - 20\right) \left(- \sqrt[4]{\left(4 \operatorname{re}{\left(y\right)} + 9\right)^{2} + 16 \left(\operatorname{im}{\left(y\right)}\right)^{2}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(4 \operatorname{im}{\left(y\right)},4 \operatorname{re}{\left(y\right)} + 9 \right)}}{2} \right)} + 8 \operatorname{re}{\left(y\right)} - 87\right)}{\left(2 \operatorname{re}{\left(y\right)} - 20\right)^{2} + 4 \left(\operatorname{im}{\left(y\right)}\right)^{2}} + \frac{\left(2 \operatorname{re}{\left(y\right)} - 20\right) \left(\sqrt[4]{\left(4 \operatorname{re}{\left(y\right)} + 9\right)^{2} + 16 \left(\operatorname{im}{\left(y\right)}\right)^{2}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(4 \operatorname{im}{\left(y\right)},4 \operatorname{re}{\left(y\right)} + 9 \right)}}{2} \right)} + 8 \operatorname{re}{\left(y\right)} - 87\right)}{\left(2 \operatorname{re}{\left(y\right)} - 20\right)^{2} + 4 \left(\operatorname{im}{\left(y\right)}\right)^{2}}$$
product
/ / / ____________________________ \ / ____________________________ \ \ / ____________________________ \ / ____________________________ \ \ / / / ____________________________ \ / ____________________________ \ \ / ____________________________ \ / ____________________________ \ \ | | | 4 / 2 2 /atan2(4*im(y), 9 + 4*re(y))\| | 4 / 2 2 /atan2(4*im(y), 9 + 4*re(y))\| | | 4 / 2 2 /atan2(4*im(y), 9 + 4*re(y))\| | 4 / 2 2 /atan2(4*im(y), 9 + 4*re(y))\| | | | | 4 / 2 2 /atan2(4*im(y), 9 + 4*re(y))\| | 4 / 2 2 /atan2(4*im(y), 9 + 4*re(y))\| | | 4 / 2 2 /atan2(4*im(y), 9 + 4*re(y))\| | 4 / 2 2 /atan2(4*im(y), 9 + 4*re(y))\| | | |(-20 + 2*re(y))*|8*im(y) - \/ (9 + 4*re(y)) + 16*im (y) *sin|---------------------------|| 2*|-87 + 8*re(y) - \/ (9 + 4*re(y)) + 16*im (y) *cos|---------------------------||*im(y)| (-20 + 2*re(y))*|-87 + 8*re(y) - \/ (9 + 4*re(y)) + 16*im (y) *cos|---------------------------|| 2*|8*im(y) - \/ (9 + 4*re(y)) + 16*im (y) *sin|---------------------------||*im(y)| | |(-20 + 2*re(y))*|8*im(y) + \/ (9 + 4*re(y)) + 16*im (y) *sin|---------------------------|| 2*|-87 + 8*re(y) + \/ (9 + 4*re(y)) + 16*im (y) *cos|---------------------------||*im(y)| (-20 + 2*re(y))*|-87 + 8*re(y) + \/ (9 + 4*re(y)) + 16*im (y) *cos|---------------------------|| 2*|8*im(y) + \/ (9 + 4*re(y)) + 16*im (y) *sin|---------------------------||*im(y)| | | \ \ 2 // \ \ 2 // | \ \ 2 // \ \ 2 // | | | \ \ 2 // \ \ 2 // | \ \ 2 // \ \ 2 // | |I*|-------------------------------------------------------------------------------------------- - ------------------------------------------------------------------------------------------| + -------------------------------------------------------------------------------------------------- + ------------------------------------------------------------------------------------|*|I*|-------------------------------------------------------------------------------------------- - ------------------------------------------------------------------------------------------| + -------------------------------------------------------------------------------------------------- + ------------------------------------------------------------------------------------| | | 2 2 2 2 | 2 2 2 2 | | | 2 2 2 2 | 2 2 2 2 | \ \ (-20 + 2*re(y)) + 4*im (y) (-20 + 2*re(y)) + 4*im (y) / (-20 + 2*re(y)) + 4*im (y) (-20 + 2*re(y)) + 4*im (y) / \ \ (-20 + 2*re(y)) + 4*im (y) (-20 + 2*re(y)) + 4*im (y) / (-20 + 2*re(y)) + 4*im (y) (-20 + 2*re(y)) + 4*im (y) /
$$\left(\frac{2 \left(- \sqrt[4]{\left(4 \operatorname{re}{\left(y\right)} + 9\right)^{2} + 16 \left(\operatorname{im}{\left(y\right)}\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(4 \operatorname{im}{\left(y\right)},4 \operatorname{re}{\left(y\right)} + 9 \right)}}{2} \right)} + 8 \operatorname{im}{\left(y\right)}\right) \operatorname{im}{\left(y\right)}}{\left(2 \operatorname{re}{\left(y\right)} - 20\right)^{2} + 4 \left(\operatorname{im}{\left(y\right)}\right)^{2}} + i \left(\frac{\left(- \sqrt[4]{\left(4 \operatorname{re}{\left(y\right)} + 9\right)^{2} + 16 \left(\operatorname{im}{\left(y\right)}\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(4 \operatorname{im}{\left(y\right)},4 \operatorname{re}{\left(y\right)} + 9 \right)}}{2} \right)} + 8 \operatorname{im}{\left(y\right)}\right) \left(2 \operatorname{re}{\left(y\right)} - 20\right)}{\left(2 \operatorname{re}{\left(y\right)} - 20\right)^{2} + 4 \left(\operatorname{im}{\left(y\right)}\right)^{2}} - \frac{2 \left(- \sqrt[4]{\left(4 \operatorname{re}{\left(y\right)} + 9\right)^{2} + 16 \left(\operatorname{im}{\left(y\right)}\right)^{2}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(4 \operatorname{im}{\left(y\right)},4 \operatorname{re}{\left(y\right)} + 9 \right)}}{2} \right)} + 8 \operatorname{re}{\left(y\right)} - 87\right) \operatorname{im}{\left(y\right)}}{\left(2 \operatorname{re}{\left(y\right)} - 20\right)^{2} + 4 \left(\operatorname{im}{\left(y\right)}\right)^{2}}\right) + \frac{\left(2 \operatorname{re}{\left(y\right)} - 20\right) \left(- \sqrt[4]{\left(4 \operatorname{re}{\left(y\right)} + 9\right)^{2} + 16 \left(\operatorname{im}{\left(y\right)}\right)^{2}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(4 \operatorname{im}{\left(y\right)},4 \operatorname{re}{\left(y\right)} + 9 \right)}}{2} \right)} + 8 \operatorname{re}{\left(y\right)} - 87\right)}{\left(2 \operatorname{re}{\left(y\right)} - 20\right)^{2} + 4 \left(\operatorname{im}{\left(y\right)}\right)^{2}}\right) \left(\frac{2 \left(\sqrt[4]{\left(4 \operatorname{re}{\left(y\right)} + 9\right)^{2} + 16 \left(\operatorname{im}{\left(y\right)}\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(4 \operatorname{im}{\left(y\right)},4 \operatorname{re}{\left(y\right)} + 9 \right)}}{2} \right)} + 8 \operatorname{im}{\left(y\right)}\right) \operatorname{im}{\left(y\right)}}{\left(2 \operatorname{re}{\left(y\right)} - 20\right)^{2} + 4 \left(\operatorname{im}{\left(y\right)}\right)^{2}} + i \left(\frac{\left(\sqrt[4]{\left(4 \operatorname{re}{\left(y\right)} + 9\right)^{2} + 16 \left(\operatorname{im}{\left(y\right)}\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(4 \operatorname{im}{\left(y\right)},4 \operatorname{re}{\left(y\right)} + 9 \right)}}{2} \right)} + 8 \operatorname{im}{\left(y\right)}\right) \left(2 \operatorname{re}{\left(y\right)} - 20\right)}{\left(2 \operatorname{re}{\left(y\right)} - 20\right)^{2} + 4 \left(\operatorname{im}{\left(y\right)}\right)^{2}} - \frac{2 \left(\sqrt[4]{\left(4 \operatorname{re}{\left(y\right)} + 9\right)^{2} + 16 \left(\operatorname{im}{\left(y\right)}\right)^{2}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(4 \operatorname{im}{\left(y\right)},4 \operatorname{re}{\left(y\right)} + 9 \right)}}{2} \right)} + 8 \operatorname{re}{\left(y\right)} - 87\right) \operatorname{im}{\left(y\right)}}{\left(2 \operatorname{re}{\left(y\right)} - 20\right)^{2} + 4 \left(\operatorname{im}{\left(y\right)}\right)^{2}}\right) + \frac{\left(2 \operatorname{re}{\left(y\right)} - 20\right) \left(\sqrt[4]{\left(4 \operatorname{re}{\left(y\right)} + 9\right)^{2} + 16 \left(\operatorname{im}{\left(y\right)}\right)^{2}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(4 \operatorname{im}{\left(y\right)},4 \operatorname{re}{\left(y\right)} + 9 \right)}}{2} \right)} + 8 \operatorname{re}{\left(y\right)} - 87\right)}{\left(2 \operatorname{re}{\left(y\right)} - 20\right)^{2} + 4 \left(\operatorname{im}{\left(y\right)}\right)^{2}}\right)$$
=
2 2
1890 - 349*re(y) + 16*im (y) + 16*re (y) + 29*I*im(y)
-----------------------------------------------------
2 2
100 + im (y) + re (y) - 20*re(y)
$$\frac{16 \left(\operatorname{re}{\left(y\right)}\right)^{2} - 349 \operatorname{re}{\left(y\right)} + 16 \left(\operatorname{im}{\left(y\right)}\right)^{2} + 29 i \operatorname{im}{\left(y\right)} + 1890}{\left(\operatorname{re}{\left(y\right)}\right)^{2} - 20 \operatorname{re}{\left(y\right)} + \left(\operatorname{im}{\left(y\right)}\right)^{2} + 100}$$
(1890 - 349*re(y) + 16*im(y)^2 + 16*re(y)^2 + 29*i*im(y))/(100 + im(y)^2 + re(y)^2 - 20*re(y))