(67.5x^2+4.5)/(x^2+8x+1)=40

Solution for (67.5x^2+4.5)/(x^2+8x+1)=40 equation:

(67.5x^2+4.5)/(x^2+8x+1)=40

We move all terms to the left:

(67.5x^2+4.5)/(x^2+8x+1)-(40)=0


Domain of the equation: (x^2+8x+1)!=0

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

x^2+8x!=-1

x∈R


We multiply all the terms by the denominator


(67.5x^2+4.5)-40*(x^2+8x+1)=0

We multiply parentheses

-40x^2+(67.5x^2+4.5)-320x-40=0

We get rid of parentheses

-40x^2+67.5x^2-320x+4.5-40=0

We add all the numbers together, and all the variables

27.5x^2-320x-35.5=0

a = 27.5; b = -320; c = -35.5;
Δ = b2-4ac
Δ = -3202-4·27.5·(-35.5)
Δ = 106305
The delta value is higher than zero, so the equation has two solutions
We use following formulas to calculate our solutions:
$x_{1}=\frac{-b-\sqrt{\Delta}}{2a}$
$x_{2}=\frac{-b+\sqrt{\Delta}}{2a}$

$x_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(-320)-\sqrt{106305}}{2*27.5}=\frac{320-\sqrt{106305}}{55}
$
$x_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(-320)+\sqrt{106305}}{2*27.5}=\frac{320+\sqrt{106305}}{55}
$