42-3x=x^2

Solution for 42-3x=x^2 equation:

42-3x=x^2

We move all terms to the left:

42-3x-(x^2)=0

determiningTheFunctionDomain
-x^2-3x+42=0

We add all the numbers together, and all the variables

-1x^2-3x+42=0

a = -1; b = -3; c = +42;
Δ = b2-4ac
Δ = -32-4·(-1)·42
Δ = 177
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{-(-3)-\sqrt{177}}{2*-1}=\frac{3-\sqrt{177}}{-2}
$
$x_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(-3)+\sqrt{177}}{2*-1}=\frac{3+\sqrt{177}}{-2}
$