(2x-1)(2x+1)=(x-5)2+307

Solution for (2x-1)(2x+1)=(x-5)2+307 equation:

(2x-1)(2x+1)=(x-5)2+307

We move all terms to the left:

(2x-1)(2x+1)-((x-5)2+307)=0

We use the square of the difference formula

4x^2-((x-5)2+307)-1=0


We calculate terms in parentheses: -((x-5)2+307), so:

(x-5)2+307

We multiply parentheses

2x-10+307

We add all the numbers together, and all the variables

2x+297

Back to the equation:

-(2x+297)


We get rid of parentheses

4x^2-2x-297-1=0

We add all the numbers together, and all the variables

4x^2-2x-298=0

a = 4; b = -2; c = -298;
Δ = b2-4ac
Δ = -22-4·4·(-298)
Δ = 4772
The delta value is higher than zero, so the equation has two solutions
We use following formulas to calculate our solutions:
$x_{1}=\frac{-b-\sqrt{\Delta}}{2a}$
$x_{2}=\frac{-b+\sqrt{\Delta}}{2a}$

The end solution:

$\sqrt{\Delta}=\sqrt{4772}=\sqrt{4*1193}=\sqrt{4}*\sqrt{1193}=2\sqrt{1193}$
$x_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(-2)-2\sqrt{1193}}{2*4}=\frac{2-2\sqrt{1193}}{8}
$
$x_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(-2)+2\sqrt{1193}}{2*4}=\frac{2+2\sqrt{1193}}{8}
$