Chapter 10, Problem 1PCT

In Problems 1-3, complete the square in the given expression. Then factor the perfect square trinomial.

1. z² + 10z

To determine

To complete: The square in $z^2+10z$ in order to make it a perfect square trinomial, also factor the perfect square trinomial.

Answer to Problem 1PCT

The perfect square trinomial is $z^2+10z+25$ and it can be factored as ${\left(z+5\right)}^2$.

Explanation of Solution

Property used:

(1) To obtain a Perfect Square Trinomial:

Multiply the coefficient of the first degree term by $\frac{1}{2}$. Square the result. That is, for $x^2+bx$, compute ${\left(\frac{1}{2}b\right)}^2$, and add the result to the expression $x^2+bx$ to obtain $x^2+bx+{\left(\frac{1}{2}b\right)}^2$.

(2) The factored form of the perfect square trinomial $x^2+bx+{\left(\frac{1}{2}b\right)}^2$ is ${\left(x+\frac{b}{2}\right)}^2$ and that of $x^2-bx+{\left(\frac{1}{2}b\right)}^2$ is ${\left(x-\frac{b}{2}\right)}^2$.

Calculation:

Note that $z^2+10z$ is in the form $x^2+bx$ where $b=10$ and $x=z$.

Substitute $b=10$ in ${\left(\frac{1}{2}b\right)}^2$ as,

$\begin{array}{c}{\left(\frac{1}{2}b\right)}^2={\left(\frac{10}{2}\right)}^2\\ =5^2\\ =25\end{array}$

Use the above property and add 25 to $z^2+10z$ as $z^2+10z+25$.

Thus, the perfect square trinomial is $z^2+10z+25$.

Notice that $z^2+10z+25$ is in the form $x^2+bx+{\left(\frac{1}{2}b\right)}^2$ which can be factored as ${\left(x+\frac{b}{2}\right)}^2$.

Substitute x with z and $b=10$ in ${\left(x+\frac{b}{2}\right)}^2$.

$\begin{array}{c}{\left(x+\frac{b}{2}\right)}^2={\left(z+\frac{10}{2}\right)}^2\\ ={\left(z+5\right)}^2\end{array}$

Therefore, the perfect square trinomial is $z^2+10z+25$ and it can be factored as ${\left(z+5\right)}^2$.