Q. 316 - Intermediate Algebra

Question

For the function, $f\left(x\right)=18x^2+15x-10$:

a. Find $x$, when $f\left(x\right)=15$.

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 $-\frac{5}{3}$ or $\frac{5}{6}$ when $f\left(x\right)=15$.

b. $(-\frac{5}{3}, 15)$ and $(\frac{5}{6}, 15)$ lie on the graph.

Part a. Find x when f ( x ) = 15 .

Substitute $f\left(x\right)=15$ in the given equation,

$$\begin{gathered} f\left(x\right)=18x^2+15x-10 \\ 15=18x^2+15x-10 \\ 18x^2+15x-25=0 \\ 18x^2+30x-15x-25=0 \\ 6x(3x+5)-5(3x+5)=0 \\ (3x+5)(6x-5)=0 \end{gathered}$$

Using Zero Product Property,

$3x+5=0$ or

$6x-5=0$.

When $3x+5=0$,

$x=-\frac{5}{3}$

When $6x-5=0$,

$x=\frac{5}{6}$.

Checking each value separately,

when $x=-\frac{5}{3}$,

$$\begin{gathered} f\left(x\right)=18x^2+15x-10 \\ f(-\frac{5}{3})=18{(-\frac{5}{3})}^2+15(-\frac{5}{3})-10 \\ f(-\frac{5}{3})=50-25-10 \\ f(-\frac{5}{3})=15 \end{gathered}$$

This is true.

When $x=\frac{5}{6}$,

$$\begin{gathered} f\left(x\right)=18x^2+15x-10 \\ f\left(\frac{5}{6}\right)=18{\left(\frac{5}{6}\right)}^2+15\left(\frac{5}{6}\right)-10 \\ f\left(\frac{5}{6}\right)=\frac{25}{2}+\frac{25}{2}-10 \\ f\left(\frac{5}{6}\right)=15 \end{gathered}$$

This is true too.

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

Since $f(-\frac{5}{3})=15$ and $f\left(\frac{5}{6}\right)=15$, the points $(-\frac{5}{3}, 15)$ and $(\frac{5}{6}, 15)$ lie on the graph of the function.