Chapter 7.7, Problem 30WE

To determine

Tofind: thequadratic equation.

Answer to Problem 30WE

The quadratic equation is $f\left(x\right)=\frac{2}{5}\left(x^2+2x-24\right)$ .

Explanation of Solution

Given information:

vertex $\left(-1,-10\right)$ and $x$ -intercepts $-6\text{ and 4}$ .

Calculation: to find the quadratic equation, first use the formula of quadratic formula with roots,

$f\left(x\right)=a\left[x^2-\left(\text{sum of roots}\right)x+\left(\text{product of roots}\right)\right]$

Now, substitute and simplify,

$\begin{array}{l}f\left(x\right)=a\left[x^2-\left(-6+4\right)x+\left(\left(-6\right)\left(4\right)\right)\right]\text{ [substitute]}\\ f\left(x\right)=a\left(x^2+2x-24\right)\text{ [simplify]}\end{array}$

Therefore vertex is $\left(-1,-10\right)$ . Since, the vertex lies on the graph of quadratic equation, so substitute its coordinates in the equation,

$\begin{array}{l}f\left(x\right)=a\left(x^2+2x-24\right)\\ -10=a\left({\left(-1\right)}^2+2\left(-1\right)-24\right)\text{ [substitute]}\\ -10=a\left(-25\right)\text{ [add and subtract]}\\ a=\frac{2}{5}\text{ [simplify]}\end{array}$

Thus, the equation is $f\left(x\right)=\frac{2}{5}\left(x^2+2x-24\right)$ .