Q 283 - Intermediate Algebra

Question

In the following exercises, solve:

$5a^2 - 26a= 24$

Step-by-Step Solution

Answer

The solutions are $a=6 and a =\frac{-4}{5}.$

Step 1. Given Information

The given quadratic equation is $5a^2 - 26a = 24.$

We have to solve the quadratic equation by factoring.

Step 2. Factor the quadratic expression

First, write the given equation in standard form.

The standard form of the quadratic equation is

$a{x}^2+bx+c=0$

Therefore,

$$\begin{gathered} 5a^2- 26a=24 \\ 5a^2-26a-24=0 \end{gathered}$$

Now, let's factor the quadratic expression

$$\begin{gathered} 5a^2-26a-24=0 \\ 5a^2-30a+4a-24=0 \\ 5a(a-6)+4(a-6)=0 \\ (5a+4)(a-6)=0 \end{gathered}$$

Step 3. Use the Zero Product Property

The equations we get are

$$\begin{gathered} 5a+4=0 \\ a-6=0 \end{gathered}$$

Thus,

$$\begin{gathered} 5a+4=0 \\ a=\frac{-4}{5} \end{gathered}$$ or $$\begin{gathered} a-6=0 \\ a=6 \end{gathered}$$

Step 4. Verify the answers

To verify the answer we will substitute the solutions separately into the original equation.

Put $a=6$ in the given equation,

$$\begin{gathered} 5{\left(6\right)}^2-26\left(6\right)=24 \\ 5\left(36\right)-156=24 \\ 180-156=24 \\ 24=24 \end{gathered}$$

This is true.

Now, put $a=\frac{-4}{5}$in the given equation

$$\begin{gathered} 5{\left(\frac{-4}{5}\right)}^2-26\left(\frac{-4}{5}\right)=24 \\ \frac{16}{5}+\frac{104}{5}=24 \\ 24=24 \end{gathered}$$

This is true.

Hence, the solutions are $a=6,\frac{-4}{5}.$