Chapter 4.2, Problem 3GP

To determine

To find: the two real numbers with a sum of 6 and a product of $-55$ .

Answer to Problem 3GP

  $x=-5\text{,}11$ .

Explanation of Solution

Given:

The sum of two real number is 6 and its product of $-55$ .

Calculation:

Represent the given condition is expressed in to standard form:

Let x be the variable for the first unknown number.

And $6-x$ for the second unknown number.

Therefore, according to the question,

  $\begin{array}{l}x\left(6-x\right)=-55\{\text{given equation}}\\ 6x-x^2=-55\left\{\text{simplify}\right\}\\ 6x-x^2+55=0\left\{\text{equate one side to 0}\right\}\end{array}$ .

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

  $\begin{array}{l}x=\frac{-b}{2a}\left\{\text{equation for symmetry}\right\}\\ x=\frac{-6}{2\left(-1\right)}\{\text{substitutes these value}}\\ x=3\end{array}$

Now write the table for the given function.

$x$	$-5$	$-1$	$3$	$7$	$11$
$f\left(x\right)$	$0$	$48$	$64$	$48$	$0$

Based on the table and the graph, the function has two zero $-5\text{and}11$ .

Hence, the solution of the given equation is $x=-5\text{,}11$