Solving Equations of the Form $ax^2 = b$

Example $1$:     $18=2x^2$
graph-y2x2
Lets start with the equation $y=2x^2$.
If we want to show all the possible solutions we would need to graphing this equation.

graph-182x2

Then we find out that $y = 18$, so the equation becomes $18=2x^2$.
If we wanted to find the solution by graphing, we would need to graph $y = 18$.

Look at where the two graphs intersect, $(-3,18)$ and $(3,18)$.
These points let us find the values
for $x$ when $y=18$.

So the solutions to the equation $18=2x^2$ are $x=-3$ and $x=3$.

Notice that graphs in the form $y = ax^2$ form parabolas, and, when we know the value of $b$, equations in the form $b = ax^2$ will have two solutions.

Example $2$:     $x^2-16 = 144$

Normally we do not solve these equations by graphing them, instead we solve them like any other equation.

Original equation                                 $x^2-16 = 144$
Add $16$ to each side                                      $x^2 = 160$
Take the square root of each side           $\sqrt{x^2} = \sqrt{160} $
Find the factor of $160$                                  $\sqrt{x^2} = \sqrt{$ 4 * 4 * 10} $
Simplify the square roots and
find the two solutions                      $x = 4\sqrt{10} $    $and$     $x = -4\sqrt{10} $

Example $3$:     $(x-4)^2 = 144$

Original equation                                               $(x-4)^2 = 144$
Take the square root of each side                 $\sqrt{(x – 4)^2} = \sqrt{$144} $

Simplify the square roots and write out the two equations
$x – 4 = 12$         $and$         $x – 4 = -12$
Solve the two equations                         $x = 16$         $and$             $x = -8$