Chapter 4.2, Problem 37PPS

To determine

To find: the two real numbers with a sum of $-8$ and a product of $-209$ by using quadratic equation.

Answer to Problem 37PPS

  $-19\text{and}11$ .

Explanation of Solution

Given:

The sum of two real number is $-8$ and its product of $-209$ .

Calculation:

Represent the given condition is expressed in to standard form:

Let x be the variable for the first unknown number.

And $-8-x$ for the second unknown number.

Therefore, according to the question,

  $\begin{array}{l}x\left(-8-x\right)=-209\{\text{given equation}}\\ -8x-x^2=-209\left\{\text{simplify}\right\}\\ -x^2-8x+209=0\left\{\text{equate one side to 0}\right\}\end{array}$ .

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

  $\begin{array}{l}x=\frac{-b}{2a}\left\{\text{equation for symmetry}\right\}\\ x=\frac{-\left(-8\right)}{2\left(-1\right)}\{\text{substitutes these value}}\\ x=\frac{-8}{2}=-4\end{array}$

Now write the table for the given function.

$x$	$-19$	$-11$	$-4$	$3$	$11$
$f\left(x\right)$	$0$	$176$	$225$	$176$	$0$

Based on the table and the graph, the function has two zero $-19\text{and}11$ .

Hence, the solution of the given equation is between $-19\text{and}11$