Q 6.50 - Intermediate Algebra

Question

Factor: $64m^2+112mn+49n^2$.

Step-by-Step Solution

Answer

Factors of $64m^2+112mn+49n^2$ is ${\left(8m+7n\right)}^2$.

Step 1. Given information.

Given an expression $64m^2+112mn+49n^2$.

Step 2. Find if first and last term are perfect squares or not.

First term is $64m^2$ which can be expressed as ${\left(8m\right)}^2$.

Last term is $49n^2$ which can be expressed as ${\left(7n\right)}^2$.

Consider $a=8m$ and $b=7n$.

So, first and last term are perfect squares.

Step 3. Check if the middle term is of the form ± 2 a b .

Here, middle term is $112mn$which can be expressed as $2\times 8m\times 7n$.

It follows that middle term is of the form .

Step 4. Write the square of the binomial.

So, $64m^2+112mn+49n^2$ is a perfect square trinomial of the form $a^2+2ab+b^2$.

Note that ${\left(a+b\right)}^2=a^2+2ab+b^2$.

Therefore, $64m^2+112mn+49n^2={\left(8m+7n\right)}^2$.

Step 5. Check by multiplying.

Verification can be obtained as follows.

$\begin{array}{rcl}{\left(8m+7n\right)}^2& =& {\left(8m\right)}^2+2\times 8m\times 7n+{\left(7n\right)}^2\\ & =& 64m^2+112mn+49n^2\end{array}$