Unit 11: Quadratic Equations and Functions

11.1: Solving Quadratic Equations

Without Leading Coefficient

Let's look at solving a quadratic equation with no leading coefficient: $x^2+bx+c=0$.

Example

Solve $x^2+5x-24=0$.

Identify $a$, $b$, and $c$.

$a=1$, $b=5$, $c=-24$.

Since $a=1$ the process is a short one.

Find factors of $c$ which add to $b$.

We need to find a pair of numbers which add to $b=5$ and multiply to $c=-24$.

Because $c$ is negative, we need one positive and one negative factor:

The pair of numbers we've found is $8$ and $-3$.

Rewrite the equation using our newly-discovered factors.

$\left(x+8\right)\left(x-3\right)=0$.

Solve each part

$\underbrace{\left(x+8\right)}_{ \begin{aligned}x+8&=0\\x&=-8\end{aligned} }\underbrace{\left(x-3\right)}_{ \begin{aligned}x-3&=0\\x&=3\end{aligned} }=0$.

We have the two answers: $x=-8$ and $x=3$.