Chapter F.4, Problem 1AYU

To determine

To fill: The statement “To complete the square of $x^2+10x$, you would ____ (add/subtract) the number ____”.

Answer to Problem 1AYU

The statement is “To complete the square of $x^2+10x$, you would add the number 25”.

Explanation of Solution

Result used:

Completing the square $x^2+bx$:

“Identify the coefficient of the first-degree term. Multiply this coefficient by $\frac{1}{2}$ and then square the result. That is, determine the value of b in $x^2+bx$ and compute ${\left(\frac{1}{2}b\right)}^2$”.

Calculation:

Compare the given $x^2+10x$ with $x^2+bx$ and obtain the value of b.

$\begin{array}{l}{x}^2+\underset{\downarrow }{b}x\\ x^2+10x\end{array}$

Thus, the value of b is 10.

Now substitute $b=10$ in ${\left(\frac{1}{2}b\right)}^2$.

${\left(\frac{1}{2}b\right)}^2={\left(\frac{1}{2}\cdot 10\right)}^2$

Obtain the complete square of $x^2+10x$ as follows.

By above result, add ${\left(\frac{1}{2}\cdot 10\right)}^2$ to $x^2+10x$ as shown below.

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

Thus, add the number 25 to the given expression in order to make the expression $x^2+10x$ becomes completed square ${\left(x+5\right)}^2$.

Hence, the statement is “To complete the square of $x^2+10x$, you would add the number 25”.