X-9=y^2-6y

Solution for X-9=y^2-6y equation:

X-9=X^2-6X

We move all terms to the left:

X-9-(X^2-6X)=0

We get rid of parentheses

-X^2+X+6X-9=0

We add all the numbers together, and all the variables

-1X^2+7X-9=0

a = -1; b = 7; c = -9;
Δ = b2-4ac
Δ = 72-4·(-1)·(-9)
Δ = 13
The delta value is higher than zero, so the equation has two solutions
We use following formulas to calculate our solutions:
$X_{1}=\frac{-b-\sqrt{\Delta}}{2a}$
$X_{2}=\frac{-b+\sqrt{\Delta}}{2a}$

$X_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(7)-\sqrt{13}}{2*-1}=\frac{-7-\sqrt{13}}{-2}
$
$X_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(7)+\sqrt{13}}{2*-1}=\frac{-7+\sqrt{13}}{-2}
$