F(1/2)=-2x^2+9x+5

Solution for F(1/2)=-2x^2+9x+5 equation:

(1/2)=-2F^2+9F+5

We move all terms to the left:

(1/2)-(-2F^2+9F+5)=0

We add all the numbers together, and all the variables

-(-2F^2+9F+5)+(+1/2)=0

We get rid of parentheses

2F^2-9F-5+1/2=0

We multiply all the terms by the denominator


2F^2*2-9F*2+1-5*2=0

We add all the numbers together, and all the variables

2F^2*2-9F*2-9=0

Wy multiply elements

4F^2-18F-9=0

a = 4; b = -18; c = -9;
Δ = b2-4ac
Δ = -182-4·4·(-9)
Δ = 468
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:

$\sqrt{\Delta}=\sqrt{468}=\sqrt{36*13}=\sqrt{36}*\sqrt{13}=6\sqrt{13}$
$F_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(-18)-6\sqrt{13}}{2*4}=\frac{18-6\sqrt{13}}{8}
$
$F_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(-18)+6\sqrt{13}}{2*4}=\frac{18+6\sqrt{13}}{8}
$