Chapter 4.2, Problem 35PPS

To determine

To find: the two real numbers with a sum of $12$ and a product of $-84$ by using quadratic equation.

Answer to Problem 35PPS

  $-5\text{ and}-4$ and $-5\text{ and}17$ .

Explanation of Solution

Given:

The sum of two real number is $12$ and its product of $-84$ .

Calculation:

Represent the given condition is expressed in to standard form:

Let x be the variable for the first unknown number.

And $12-x$ for the second unknown number.

Therefore, according to the question,

  $\begin{array}{l}x\left(12-x\right)=-84\{\text{given equation}}\\ 12x-x^2=-84\left\{\text{simplify}\right\}\\ -x^2+12x+84=0\left\{\text{equate one side to 0}\right\}\end{array}$ .

Now, graph the related quadratic function to show that x -intercept of the graph. Therefore,

  

Looking at the graph, the parabola crosses the x -axis, therefore, the equation has the solution.

The axis of symmetric is

  $\begin{array}{l}x=\frac{-b}{2a}\left\{\text{equation for symmetry}\right\}\\ x=\frac{-12}{2\left(-1\right)}\{\text{substitutes these value}}\\ x=\frac{12}{2}=6\end{array}$

Now write the table for the given function.

$x$	$-5$	$-4$	$6$	$16$	$17$
$f\left(x\right)$	$-1$	$20$	$120$	$20$	$-1$

Based on the table shown, the x -intercept indicate that one solution is between $-5\text{ and}-4$ and the other solution is $-5\text{ and}17$ .

Hence, the solution of the given equation is between $-5\text{ and}-4$ and $-5\text{ and}17$ .