x(2x-3)=52

Solution for x(2x-3)=52 equation:

x(2x-3)=52

We move all terms to the left:

x(2x-3)-(52)=0

We multiply parentheses

2x^2-3x-52=0

a = 2; b = -3; c = -52;
Δ = b2-4ac
Δ = -32-4·2·(-52)
Δ = 425
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:

$\sqrt{\Delta}=\sqrt{425}=\sqrt{25*17}=\sqrt{25}*\sqrt{17}=5\sqrt{17}$
$x_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(-3)-5\sqrt{17}}{2*2}=\frac{3-5\sqrt{17}}{4}
$
$x_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(-3)+5\sqrt{17}}{2*2}=\frac{3+5\sqrt{17}}{4}
$