Chapter 14.8, Problem 21HP

To determine

To calculate: Thepositive integer value to complete the trinomial $x^2+6x+\_$ so that the trinomial can be factored.

Answer to Problem 21HP

The positive integer value to complete the trinomial $x^2+6x+\_$ are 9, 5.

Explanation of Solution

Given Information:

The trinomial is $x^2+6x+\_$ .

Calculation:

The given trinomial is $x^2+6x+\_$ .

Consider $a$ be the positive integer value to complete the trinomial $x^2+6x+\_$ .

Thus, the trinomial is $x^2+6x+a$ .

Now, find the factors of the expression $x^2+6x+a$ .

Split $6x$ into two parts such that their product is equal to the product of $x^2$ and $a$ .

Suppose $6x$ is splits into $3x+3x$ or $5x+x$ .

  $\begin{array}{c}{x}^2+6x+a=x^2+3x+3x+a\\ =x\left(x+3\right)+3\left(x+\frac{a}{3}\right)\end{array}$

Now, take common $x+3$ , $\frac{a}{3}$ must be equal to 3. Thus,

  $\begin{array}{l}\frac{a}{3}=3\\ a=9\end{array}$

Substitute 9 for $a$ in the expression $x^2+6x+a$ .

  $x^2+6x+a=x^2+6x+9$

Thus, the expression is $x^2+6x+9$ .

  $\begin{array}{c}{x}^2+6x+a=x^2+5x+x+a\\ =x\left(x+5\right)+1\left(x+a\right)\end{array}$

Now, take common $x+5$ , $a$ must be equal to 5. Thus,

  $a=5$

Substitute 5 for $a$ in the expression $x^2+6x+a$ .

  $x^2+6x+a=x^2+6x+5$

Thus, the expression is $x^2+6x+9$ .

Hence, the positive integer value to complete the trinomial $x^2+6x+\_$ are 9, 5.