Q. 295 - Intermediate Algebra

Question

In the following exercises, solve:

$\left(x+6\right)\left(x-3\right)=-8$

Step-by-Step Solution

Answer

On solving the equation, we get $x=-5,2$.

Step 1. Given information.

Consider the given equation,

$\left(x+6\right)\left(x-3\right)=-8$

Write the quadratic equation in standard form,

$$\begin{gathered} x^2-3x+6x-18=-8 \\ {x}^2+3x-10=0 \end{gathered}$$

Step 2. Factor the quadratic expression.

On factoring, we get,

$$\begin{gathered} x^2+5x-2x-10=0 \\ x\left(x+5\right)-2\left(x+5\right)=0 \\ \left(x+5\right)\left(x-2\right)=0 \end{gathered}$$

Use the zero product property,

$$\begin{gathered} x+5=0 \\ x=-5 \end{gathered}$$

Then,

$$\begin{gathered} x-2=0 \\ x=2 \end{gathered}$$

Step 3. Check the answers.

Substitute $x=-5$ in the given equation,

$$\begin{gathered} \left(-5+6\right)\left(-5-3\right)=-8 \\ \left(1\right)\left(-8\right)=-8 \\ -8=-8 \end{gathered}$$

This is true.

Substitute $x=2$ in the given equation,

$$\begin{gathered} \left(2+6\right)\left(2-3\right)=-8 \\ \left(8\right)\left(-1\right)=-8 \\ -8=-8 \end{gathered}$$

This is also true.

Hence, the answer is $x=-5,2$.