42-x=x^2

Solution for 42-x=x^2 equation:

42-x=x^2

We move all terms to the left:

42-x-(x^2)=0

determiningTheFunctionDomain
-x^2-x+42=0

We add all the numbers together, and all the variables

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

a = -1; b = -1; c = +42;
Δ = b2-4ac
Δ = -12-4·(-1)·42
Δ = 169
The delta value is higher than zero, so the equation has two solutions
We use following formulas to calculate our solutions:
$x_{1}=\frac{-b-\sqrt{\Delta}}{2a}$
$x_{2}=\frac{-b+\sqrt{\Delta}}{2a}$

$\sqrt{\Delta}=\sqrt{169}=13$
$x_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(-1)-13}{2*-1}=\frac{-12}{-2}
=+6
$
$x_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(-1)+13}{2*-1}=\frac{14}{-2}
=-7
$