65x-x^2=424

Solution for 65x-x^2=424 equation:

65x-x^2=424

We move all terms to the left:

65x-x^2-(424)=0

We add all the numbers together, and all the variables

-1x^2+65x-424=0

a = -1; b = 65; c = -424;
Δ = b2-4ac
Δ = 652-4·(-1)·(-424)
Δ = 2529
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:

$\sqrt{\Delta}=\sqrt{2529}=\sqrt{9*281}=\sqrt{9}*\sqrt{281}=3\sqrt{281}$
$x_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(65)-3\sqrt{281}}{2*-1}=\frac{-65-3\sqrt{281}}{-2}
$
$x_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(65)+3\sqrt{281}}{2*-1}=\frac{-65+3\sqrt{281}}{-2}
$