3x+9=x*x-20

Solution for 3x+9=x*x-20 equation:

3x+9=x*x-20

We move all terms to the left:

3x+9-(x*x-20)=0

We get rid of parentheses

3x-x*x+20+9=0

We add all the numbers together, and all the variables

3x-x*x+29=0

Wy multiply elements

-1x^2+3x+29=0

a = -1; b = 3; c = +29;
Δ = b2-4ac
Δ = 32-4·(-1)·29
Δ = 125
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:

$\sqrt{\Delta}=\sqrt{125}=\sqrt{25*5}=\sqrt{25}*\sqrt{5}=5\sqrt{5}$
$x_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(3)-5\sqrt{5}}{2*-1}=\frac{-3-5\sqrt{5}}{-2}
$
$x_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(3)+5\sqrt{5}}{2*-1}=\frac{-3+5\sqrt{5}}{-2}
$