Chapter 8, Problem 29GR

To determine

Which of the given product is a perfect- square trinomial when multiplied.

Answer to Problem 29GR

The product $\left(x+4\right)\left(x+4\right)$ is a perfect square.

Explanation of Solution

Given information:

The given products are (a) $\left(x+4\right)\left(x+7\right)$ , and (b) $\left(x+4\right)\left(x+4\right)$

Formulaused:

A perfect square trinomial is given as $a^2+2ab+b^2={\left(a+b\right)}^2$ .

Calculations:

The products are $p_1=\left(x+4\right)\left(x+7\right)$ and $p_2=\left(x+4\right)\left(x+4\right)$

  $p_1=\left(x+4\right)\left(x+7\right)$

  $\Rightarrow p_1=x^2+7x+4x+28$

  $\Rightarrow p_1=x^2+11x+28$

  $\Rightarrow p_1=x^2+2\cdot x\cdot \frac{11}{2}+{\left(\frac{11}{2}\right)}^2-{\left(\frac{11}{2}\right)}^2+28$

  $\Rightarrow p_1={\left(x+\frac{11}{2}\right)}^2+\left[28-{\left(\frac{11}{2}\right)}^2\right]$

Since $\left[28-{\left(\frac{11}{2}\right)}^2\right]\ne 0$ , hence $p_1=\left(x+4\right)\left(x+7\right)$ is not a perfect square.

Now,the products are $p_2=\left(x+4\right)\left(x+4\right)={\left(x+4\right)}^2$

Thus, the product $p_2=\left(x+4\right)\left(x+4\right)$ is a perfect square.

Conclusion:

The product $\left(x+4\right)\left(x+4\right)$ is a perfect square.