Mister Exam

# Express x in terms of y where ﻿60x^2y+75x^2y^2+120xy^260x2y+75x2y2+120xy2﻿при﻿x=-0,5x=−0,5﻿,﻿y=0,4y=0,4﻿.

A equation with variable:

#### Numerical solution:

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    2         2  2          260                          2           -x
60*x *y + 75*x *y  + 120*x*y   *x2*y + 75*x2*y2 + 120*x*y *pi*pi*x = ---
2 
$$x \pi \pi 120 x y^{2} + \left(75 x_{2} y_{2} + \left(y x_{2} \cdot 120 x y^{260} + \left(60 x^{2} y + 75 x^{2} y^{2}\right)\right)\right) = - \frac{x}{2}$$
Detail solution
Move right part of the equation to
left part with negative sign.

The equation is transformed from
$$x \pi \pi 120 x y^{2} + \left(75 x_{2} y_{2} + \left(y x_{2} \cdot 120 x y^{260} + \left(60 x^{2} y + 75 x^{2} y^{2}\right)\right)\right) = - \frac{x}{2}$$
to
$$\frac{x}{2} + \left(x \pi \pi 120 x y^{2} + \left(75 x_{2} y_{2} + \left(y x_{2} \cdot 120 x y^{260} + \left(60 x^{2} y + 75 x^{2} y^{2}\right)\right)\right)\right) = 0$$
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
$$a = 75 y^{2} + 120 \pi^{2} y^{2} + 60 y$$
$$b = 120 x_{2} y^{261} + \frac{1}{2}$$
$$c = 75 x_{2} y_{2}$$
, then
D = b^2 - 4 * a * c =

(1/2 + 120*x2*y^261)^2 - 4 * (60*y + 75*y^2 + 120*pi^2*y^2) * (75*x2*y2) = (1/2 + 120*x2*y^261)^2 - 75*x2*y2*(240*y + 300*y^2 + 480*pi^2*y^2)

The equation has two roots.
x1 = (-b + sqrt(D)) / (2*a)

x2 = (-b - sqrt(D)) / (2*a)

or
$$x_{1} = \frac{- 120 x_{2} y^{261} + \sqrt{- 75 x_{2} y_{2} \left(300 y^{2} + 480 \pi^{2} y^{2} + 240 y\right) + \left(120 x_{2} y^{261} + \frac{1}{2}\right)^{2}} - \frac{1}{2}}{150 y^{2} + 240 \pi^{2} y^{2} + 120 y}$$
$$x_{2} = \frac{- 120 x_{2} y^{261} - \sqrt{- 75 x_{2} y_{2} \left(300 y^{2} + 480 \pi^{2} y^{2} + 240 y\right) + \left(120 x_{2} y^{261} + \frac{1}{2}\right)^{2}} - \frac{1}{2}}{150 y^{2} + 240 \pi^{2} y^{2} + 120 y}$$
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