2x^2-9x-65=x^2-4

Solution for 2x^2-9x-65=x^2-4 equation:

2x^2-9x-65=x^2-4

We move all terms to the left:

2x^2-9x-65-(x^2-4)=0

We get rid of parentheses

2x^2-x^2-9x+4-65=0

We add all the numbers together, and all the variables

x^2-9x-61=0

a = 1; b = -9; c = -61;
Δ = b2-4ac
Δ = -92-4·1·(-61)
Δ = 325
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:

$\sqrt{\Delta}=\sqrt{325}=\sqrt{25*13}=\sqrt{25}*\sqrt{13}=5\sqrt{13}$
$x_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(-9)-5\sqrt{13}}{2*1}=\frac{9-5\sqrt{13}}{2}
$
$x_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(-9)+5\sqrt{13}}{2*1}=\frac{9+5\sqrt{13}}{2}
$