P(1/2)=2x^2-9x-8

Solution for P(1/2)=2x^2-9x-8 equation:

(1/2)=2P^2-9P-8

We move all terms to the left:

(1/2)-(2P^2-9P-8)=0

We add all the numbers together, and all the variables

-(2P^2-9P-8)+(+1/2)=0

We get rid of parentheses

-2P^2+9P+8+1/2=0

We multiply all the terms by the denominator


-2P^2*2+9P*2+1+8*2=0

We add all the numbers together, and all the variables

-2P^2*2+9P*2+17=0

Wy multiply elements

-4P^2+18P+17=0

a = -4; b = 18; c = +17;
Δ = b2-4ac
Δ = 182-4·(-4)·17
Δ = 596
The delta value is higher than zero, so the equation has two solutions
We use following formulas to calculate our solutions:
$P_{1}=\frac{-b-\sqrt{\Delta}}{2a}$
$P_{2}=\frac{-b+\sqrt{\Delta}}{2a}$

The end solution:

$\sqrt{\Delta}=\sqrt{596}=\sqrt{4*149}=\sqrt{4}*\sqrt{149}=2\sqrt{149}$
$P_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(18)-2\sqrt{149}}{2*-4}=\frac{-18-2\sqrt{149}}{-8}
$
$P_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(18)+2\sqrt{149}}{2*-4}=\frac{-18+2\sqrt{149}}{-8}
$