(x+5)(x-7)=42

Solution for (x+5)(x-7)=42 equation:

(x+5)(x-7)=42

We move all terms to the left:

(x+5)(x-7)-(42)=0

We multiply parentheses ..

(+x^2-7x+5x-35)-42=0

We get rid of parentheses

x^2-7x+5x-35-42=0

We add all the numbers together, and all the variables

x^2-2x-77=0

a = 1; b = -2; c = -77;
Δ = b2-4ac
Δ = -22-4·1·(-77)
Δ = 312
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:

$\sqrt{\Delta}=\sqrt{312}=\sqrt{4*78}=\sqrt{4}*\sqrt{78}=2\sqrt{78}$
$x_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(-2)-2\sqrt{78}}{2*1}=\frac{2-2\sqrt{78}}{2}
$
$x_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(-2)+2\sqrt{78}}{2*1}=\frac{2+2\sqrt{78}}{2}
$