((7x^2+3x)/5x+9)=12

Solution for ((7x^2+3x)/5x+9)=12 equation:

((7x^2+3x)/5x+9)=12

We move all terms to the left:

((7x^2+3x)/5x+9)-(12)=0


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

x∈R


We multiply all the terms by the denominator


((7x^2+3x)-12*5x+9)=0


We calculate terms in parentheses: +((7x^2+3x)-12*5x+9), so:

(7x^2+3x)-12*5x+9

Wy multiply elements

(7x^2+3x)-60x+9

We get rid of parentheses

7x^2+3x-60x+9

We add all the numbers together, and all the variables

7x^2-57x+9

Back to the equation:

+(7x^2-57x+9)


We get rid of parentheses

7x^2-57x+9=0

a = 7; b = -57; c = +9;
Δ = b2-4ac
Δ = -572-4·7·9
Δ = 2997
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:

$\sqrt{\Delta}=\sqrt{2997}=\sqrt{81*37}=\sqrt{81}*\sqrt{37}=9\sqrt{37}$
$x_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(-57)-9\sqrt{37}}{2*7}=\frac{57-9\sqrt{37}}{14}
$
$x_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(-57)+9\sqrt{37}}{2*7}=\frac{57+9\sqrt{37}}{14}
$