(x+6)/(x^2-36)=4/5

Solution for (x+6)/(x^2-36)=4/5 equation:

(x+6)/(x^2-36)=4/5

We move all terms to the left:

(x+6)/(x^2-36)-(4/5)=0


Domain of the equation: (x^2-36)!=0

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

x^2!=36

x^2!=36/

x^2!=√1/0

x!=1

x∈R


We add all the numbers together, and all the variables

(x+6)/(x^2-36)-(+4/5)=0

We get rid of parentheses

(x+6)/(x^2-36)-4/5=0

We calculate fractions


(-4x^2+144)/(5x^2-180)+(5x+30)/(5x^2-180)=0

We multiply all the terms by the denominator


(-4x^2+144)+(5x+30)=0

We get rid of parentheses

-4x^2+5x+144+30=0

We add all the numbers together, and all the variables

-4x^2+5x+174=0

a = -4; b = 5; c = +174;
Δ = b2-4ac
Δ = 52-4·(-4)·174
Δ = 2809
The delta value is higher than zero, so the equation has two solutions
We use following formulas to calculate our solutions:
$x_{1}=\frac{-b-\sqrt{\Delta}}{2a}$
$x_{2}=\frac{-b+\sqrt{\Delta}}{2a}$

$\sqrt{\Delta}=\sqrt{2809}=53$
$x_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(5)-53}{2*-4}=\frac{-58}{-8}
=7+1/4
$
$x_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(5)+53}{2*-4}=\frac{48}{-8}
=-6
$