F(x)=((x+3)(x-1))/((x-2)(x+2))

Solution for F(x)=((x+3)(x-1))/((x-2)(x+2)) equation:

(F)=((F+3)(F-1))/((F-2)(F+2))

We move all terms to the left:

(F)-(((F+3)(F-1))/((F-2)(F+2)))=0


Domain of the equation: ((F-2)(F+2)))!=0

F∈R


We multiply parentheses ..

-(((+F^2-1F+3F-3))/((F-2)(F+2)))+F=0

We multiply all the terms by the denominator


-(((+F^2-1F+3F-3))+F*((F-2)(F+2)))=0


We calculate terms in parentheses: -(((+F^2-1F+3F-3))+F*((F-2)(F+2))), so:

((+F^2-1F+3F-3))+F*((F-2)(F+2))

We use the square of the difference formula

((+F^2-1F+3F-3))+F^2-4


We calculate terms in parentheses: +((+F^2-1F+3F-3)), so:

(+F^2-1F+3F-3)

We get rid of parentheses

F^2-1F+3F-3

We add all the numbers together, and all the variables

F^2+2F-3

Back to the equation:

+(F^2+2F-3)


We get rid of parentheses

F^2+F^2+2F-3-4

We add all the numbers together, and all the variables

2F^2+2F-7

Back to the equation:

-(2F^2+2F-7)


We get rid of parentheses

-2F^2-2F+7=0

a = -2; b = -2; c = +7;
Δ = b2-4ac
Δ = -22-4·(-2)·7
Δ = 60
The delta value is higher than zero, so the equation has two solutions
We use following formulas to calculate our solutions:
$F_{1}=\frac{-b-\sqrt{\Delta}}{2a}$
$F_{2}=\frac{-b+\sqrt{\Delta}}{2a}$

The end solution:

$\sqrt{\Delta}=\sqrt{60}=\sqrt{4*15}=\sqrt{4}*\sqrt{15}=2\sqrt{15}$
$F_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(-2)-2\sqrt{15}}{2*-2}=\frac{2-2\sqrt{15}}{-4}
$
$F_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(-2)+2\sqrt{15}}{2*-2}=\frac{2+2\sqrt{15}}{-4}
$