x^2/(4-5x+x^2)=16

Solution for x^2/(4-5x+x^2)=16 equation:

x^2/(4-5x+x^2)=16

We move all terms to the left:

x^2/(4-5x+x^2)-(16)=0


Domain of the equation: (4-5x+x^2)!=0

We move all terms containing x to the left, all other terms to the right

x^2-5x!=-4

x∈R


We multiply all the terms by the denominator


x^2-16*(4-5x+x^2)=0

We multiply parentheses

x^2-16x^2+80x-64=0

We add all the numbers together, and all the variables

-15x^2+80x-64=0

a = -15; b = 80; c = -64;
Δ = b2-4ac
Δ = 802-4·(-15)·(-64)
Δ = 2560
The delta value is higher than zero, so the equation has two solutions
We use following formulas to calculate our solutions:
$x_{1}=\frac{-b-\sqrt{\Delta}}{2a}$
$x_{2}=\frac{-b+\sqrt{\Delta}}{2a}$

The end solution:

$\sqrt{\Delta}=\sqrt{2560}=\sqrt{256*10}=\sqrt{256}*\sqrt{10}=16\sqrt{10}$
$x_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(80)-16\sqrt{10}}{2*-15}=\frac{-80-16\sqrt{10}}{-30}
$
$x_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(80)+16\sqrt{10}}{2*-15}=\frac{-80+16\sqrt{10}}{-30}
$