((7*(x*x))+3x)/(5x+9)=12

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

((7(x*x))+3x)/(5x+9)=12

We move all terms to the left:

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


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

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

5x!=-9

x!=-9/5

x!=-1+4/5

x∈R


We add all the numbers together, and all the variables

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

We multiply all the terms by the denominator


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


We calculate terms in parentheses: +((7(+x*x))+3x), so:

(7(+x*x))+3x

We add all the numbers together, and all the variables

3x+(7(+x*x))


We calculate terms in parentheses: +(7(+x*x)), so:

7(+x*x)

We multiply parentheses

7x^2

Back to the equation:

+(7x^2)


determiningTheFunctionDomain
7x^2+3x

Back to the equation:

+(7x^2+3x)


We multiply parentheses

(7x^2+3x)-60x-108=0

We get rid of parentheses

7x^2+3x-60x-108=0

We add all the numbers together, and all the variables

7x^2-57x-108=0

a = 7; b = -57; c = -108;
Δ = b2-4ac
Δ = -572-4·7·(-108)
Δ = 6273
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:

$\sqrt{\Delta}=\sqrt{6273}=\sqrt{9*697}=\sqrt{9}*\sqrt{697}=3\sqrt{697}$
$x_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(-57)-3\sqrt{697}}{2*7}=\frac{57-3\sqrt{697}}{14}
$
$x_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(-57)+3\sqrt{697}}{2*7}=\frac{57+3\sqrt{697}}{14}
$