(x-5)(x-7)=13

Solution for (x-5)(x-7)=13 equation:

(x-5)(x-7)=13

We move all terms to the left:

(x-5)(x-7)-(13)=0

We multiply parentheses ..

(+x^2-7x-5x+35)-13=0

We get rid of parentheses

x^2-7x-5x+35-13=0

We add all the numbers together, and all the variables

x^2-12x+22=0

a = 1; b = -12; c = +22;
Δ = b2-4ac
Δ = -122-4·1·22
Δ = 56
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:

$\sqrt{\Delta}=\sqrt{56}=\sqrt{4*14}=\sqrt{4}*\sqrt{14}=2\sqrt{14}$
$x_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(-12)-2\sqrt{14}}{2*1}=\frac{12-2\sqrt{14}}{2}
$
$x_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(-12)+2\sqrt{14}}{2*1}=\frac{12+2\sqrt{14}}{2}
$