Question

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

when $f\left(x\right)=-8$.(b) Use this information to find two

points that lie on the graph of the function.

Short answer

(a) The $x$values for $f\left(x\right)=-8$are $x=-7,-4$

(b) Two points that lie on the graph of function are$(-7,-8),(-4,-8)$

Step-by-step solution

Step 1. Given

The function is $f\left(x\right)=x^2+11x+20$

(a) We have to find when $f\left(x\right)=-8$

(b) We also have to find two points that lie on the graph of the function.

Step 2. Find values of x when f(x)=-8

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

So,

$x=-7,-4$

Step 3. Check the value of f(x) at x=-7,-4

Substitute $x=-7$in $f\left(x\right)$,

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

Substitute $x=-4$in $f\left(x\right)$

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

Step 4. Find two points that lie on the graph of the function. 

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