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

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

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

We move all terms to the left:

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

We multiply parentheses ..

-(2(+6F^2-9F+2F-3))+F=0


We calculate terms in parentheses: -(2(+6F^2-9F+2F-3)), so:

2(+6F^2-9F+2F-3)

We multiply parentheses

12F^2-18F+4F-6

We add all the numbers together, and all the variables

12F^2-14F-6

Back to the equation:

-(12F^2-14F-6)


We add all the numbers together, and all the variables

F-(12F^2-14F-6)=0

We get rid of parentheses

-12F^2+F+14F+6=0

We add all the numbers together, and all the variables

-12F^2+15F+6=0

a = -12; b = 15; c = +6;
Δ = b2-4ac
Δ = 152-4·(-12)·6
Δ = 513
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:

$\sqrt{\Delta}=\sqrt{513}=\sqrt{9*57}=\sqrt{9}*\sqrt{57}=3\sqrt{57}$
$F_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(15)-3\sqrt{57}}{2*-12}=\frac{-15-3\sqrt{57}}{-24}
$
$F_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(15)+3\sqrt{57}}{2*-12}=\frac{-15+3\sqrt{57}}{-24}
$