51-6x-x^2=0

Solution for 51-6x-x^2=0 equation:

51-6x-x^2=0

We add all the numbers together, and all the variables

-1x^2-6x+51=0

a = -1; b = -6; c = +51;
Δ = b2-4ac
Δ = -62-4·(-1)·51
Δ = 240
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:

$\sqrt{\Delta}=\sqrt{240}=\sqrt{16*15}=\sqrt{16}*\sqrt{15}=4\sqrt{15}$
$x_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(-6)-4\sqrt{15}}{2*-1}=\frac{6-4\sqrt{15}}{-2}
$
$x_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(-6)+4\sqrt{15}}{2*-1}=\frac{6+4\sqrt{15}}{-2}
$