Q 287 - Intermediate Algebra

Question

In the following exercises, solve:

$7a^2 + 14a = 7a$

Step-by-Step Solution

Answer

The solutions are $a=0,-1.$

Step 1. Given Information

The given quadratic equation is $7a^2+ 14a = 7a.$

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,

$$\begin{gathered} 7a^2+14a=7a \\ 7a^2+14a-7a=0 \\ 7a^2+7a=0 \\ 7a(a+1)=0 \end{gathered}$$

Step 3. Use the Zero Product Property

The equations we get are

$$\begin{gathered} 7a=0 \\ a+1=0 \end{gathered}$$

Thus,

$$\begin{gathered} 7a=0 \\ a=0 \end{gathered}$$ or $$\begin{gathered} a+1=0 \\ a=-1 \end{gathered}$$

Step 4. Verify the answers

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

Put $a=0$ in the given equation,

$$\begin{gathered} 7{\left(0\right)}^2+14\left(0\right)=7\left(0\right) \\ 0+0=0 \\ 0=0 \end{gathered}$$

It is true.

Now, put $a=-1$ in the given equation

$$\begin{gathered} 7{(-1)}^2+14(-1)=7(-1) \\ 7-14=-7 \\ -7=-7 \end{gathered}$$

It is also true.

Hence, the solutions are $a=0,-1.$