P(1/2)=2x^2-3x-6

Solution for P(1/2)=2x^2-3x-6 equation:

(1/2)=2P^2-3P-6

We move all terms to the left:

(1/2)-(2P^2-3P-6)=0

We add all the numbers together, and all the variables

-(2P^2-3P-6)+(+1/2)=0

We get rid of parentheses

-2P^2+3P+6+1/2=0

We multiply all the terms by the denominator


-2P^2*2+3P*2+1+6*2=0

We add all the numbers together, and all the variables

-2P^2*2+3P*2+13=0

Wy multiply elements

-4P^2+6P+13=0

a = -4; b = 6; c = +13;
Δ = b2-4ac
Δ = 62-4·(-4)·13
Δ = 244
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:

$\sqrt{\Delta}=\sqrt{244}=\sqrt{4*61}=\sqrt{4}*\sqrt{61}=2\sqrt{61}$
$P_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(6)-2\sqrt{61}}{2*-4}=\frac{-6-2\sqrt{61}}{-8}
$
$P_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(6)+2\sqrt{61}}{2*-4}=\frac{-6+2\sqrt{61}}{-8}
$