Q. 463 - Intermediate Algebra

Question

For the function, $f\left(x\right)=x^2-7x+5$, find

(a) when $f\left(x\right)=-7$

(b) Use this information to find two

points that lie on the graph of the function.

Step-by-Step Solution

Answer

(a) The values when $f\left(x\right)=-7$ are $x=3 or 4$.

(b) Two points that lie on the graph are $(4,-7),(3,-7)$.

Step 1. Given and explanation

We have $f\left(x\right)=x^2-7x+5$.

To find the values when $f\left(x\right)=-7$, substitute this value in the equation and then simplify the equation.

Put the resulting values in the equation to find two points that lie on the graph.

Step 2. Simplifying the trinomial.

(a) We have $f\left(x\right)=x^2-7x+5$.

Putting the value of $f\left(x\right)=-7$ gives us,

$-7=x^2-7x+5$

Adding $7$ both sides,

$$\begin{gathered} -7+7=x^2-7x+5+7 \\ {x}^2-7x+12=0 \end{gathered}$$

Since $-7x=-4x-3x$ and
$(-4)(-3)=12$ , we can write this as,

$$\begin{gathered} x^2-7x+12=0 \\ {x}^2-4x-3x+12=0 \\ x(x-4)-3(x-4)=0 \\ (x-3)(x-4)=0 \end{gathered}$$

Using the zero product rule gives us,

1. $$\begin{gathered} x-3=0 \\ x=3 \end{gathered}$$
   
2. $$\begin{gathered} x-4=0 \\ x=4 \end{gathered}$$
   

This shows that values are $3,4$.

Step 3. Substituting values in functions.

We have $f\left(x\right)=x^2-7x+5$.

Putting $x=3,4$ gives us,

1. When $x=3$
   $$\begin{gathered} ={\left(3\right)}^2-7\left(3\right)+5 \\ =9-21+5 \\ =14-21 \\ =-7 \end{gathered}$$
   
2. When $x=4$
   $$\begin{gathered} ={\left(4\right)}^2-7\left(4\right)+5 \\ =16-28+5 \\ =21-28 \\ =-7 \end{gathered}$$
   

Step 4. Solution for part (b)

As we know that $f\left(3\right)=-4$ and
$f\left(4\right)=-7$, So the points that lie on the graph are $(3,-7),(4,-7)$.