(((3x^2)+30x-69)/(5x-9))=8

Solution for (((3x^2)+30x-69)/(5x-9))=8 equation:

(((3x^2)+30x-69)/(5x-9))=8

We move all terms to the left:

(((3x^2)+30x-69)/(5x-9))-(8)=0


Domain of the equation: (5x-9))!=0

x∈R


We multiply all the terms by the denominator


((3x^2+30x-69)-8*(5x-9))=0


We calculate terms in parentheses: +((3x^2+30x-69)-8*(5x-9)), so:

(3x^2+30x-69)-8*(5x-9)

We multiply parentheses

(3x^2+30x-69)-40x+72

We get rid of parentheses

3x^2+30x-40x-69+72

We add all the numbers together, and all the variables

3x^2-10x+3

Back to the equation:

+(3x^2-10x+3)


We get rid of parentheses

3x^2-10x+3=0

a = 3; b = -10; c = +3;
Δ = b2-4ac
Δ = -102-4·3·3
Δ = 64
The delta value is higher than zero, so the equation has two solutions
We use following formulas to calculate our solutions:
$x_{1}=\frac{-b-\sqrt{\Delta}}{2a}$
$x_{2}=\frac{-b+\sqrt{\Delta}}{2a}$

$\sqrt{\Delta}=\sqrt{64}=8$
$x_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(-10)-8}{2*3}=\frac{2}{6}
=1/3
$
$x_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(-10)+8}{2*3}=\frac{18}{6}
=3
$