Chapter 4.2, Problem 55HP

To determine

To write: the quadratic function in the given condition.

Answer to Problem 55HP

  $-5\left(x+2\right)\left(x-8\right)$ .

Explanation of Solution

Given:

The quadratic equation has maximum at $\left(3,125\right)$ has a root at $-2\text{ and 8}$ .

Calculation:

The general form of factorized quadratic equation, with roots $s\text{ and }t$ is given by $a\left(x-s\right)\left(x-t\right)$ .

Therefore, the quadratic equation is:

  $\begin{array}{l}a\left(x-8\right)\left(x-\left(-2\right)\right)\text{ {substitute these value}}\\ =a\left(x-8\right)\left(x+2\right)\\ \{\text{at}x=3,\text{ the function is 125, plug the value in above equation} then,}\\ a\left(3-8\right)\left(3+2\right)=125\\ a\left(-5\right)\left(5\right)=125\\ \therefore a=-5\end{array}$

Thus, the required equation is $-5\left(x+2\right)\left(x-8\right)$ .

  $\Rightarrow -6-\left(-11\right)=5$

Hence, the quadratic function has maximum at $\left(3,125\right)$ is $-5\left(x+2\right)\left(x-8\right)$ .