Solving Quadratic Equations
- Solving $ax^2 = b$
Example $1$: $18=2x^2$
Lets start with the equation $y=2x^2$.
If we want to show all the possible solutions we would need to graphing this equation.
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 $x = 16$ $and$ $x = -8$
