Q. 314 - Intermediate Algebra

Question

For the function, $f\left(x\right)=x^2+11x+20$:

a. find $x$, when $f\left(x\right)=-8$.

b. Use this information to find two points that lie on the graph of the function.

Step-by-Step Solution

Answer

a. The value of $x$ is $-4$ or $-7$ when $f\left(x\right)=-8$.

b. $(-4, -8)$ and $(-7, -8)$ lie on the graph of the function.

Part a. Find x when f ( x ) = - 8 .

Substitute $f\left(x\right)=-8$ in the given equation,

$$\begin{gathered} f\left(x\right)=x^2+11x+20 \\ -8=x^2+11x+20 \\ {x}^2+11x+28=0 \\ {x}^2+4x+7x+28=0 \\ x(x+4)+7(x+4)=0 \\ (x+4)(x+7)=0 \end{gathered}$$

Using Zero Product Property,

$x+4=0$or

$x+7=0$.

When $x+4=0$,

$x=-4$.

When $x+7=0$,

$x=-7$.

Checking each value separately,

when $x=-4$,

$$\begin{gathered} f\left(x\right)=x^2+11x+20 \\ f(-4)={\left(-4\right)}^2+11(-4)+20 \\ f(-4)= 16-44+20 \\ f(-4)=-8 \end{gathered}$$

This is true.

When $x=-7$,

$$\begin{gathered} f\left(x\right)=x^2+11x+20 \\ f(-7)={(-7)}^2+11(-7)+20 \\ f(-7)= 49-77+20 \\ f(-7)=-8 \end{gathered}$$

This is true too.

Part b. Find points that lie on the graph of the function

Since $f(-4)=-8$ and $f(-7)=-8$, the points $(-4, -8)$ and $(-7, -8)$ lie on the graph of the given function.