Q 6.49TI. - Intermediate Algebra

Question

Factor: $49x^2+84xy+36y^2$.

Step-by-Step Solution

Answer

Factors of $49x^2+84xy+36y^2$ is ${\left(7x+6y\right)}^2$.

Step 1. Given information.

Given an expression $49x^2+84xy+36y^2$.

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

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

Last term is $36y^2$ which can be expressed as ${\left(6y\right)}^2$.

Consider $a=7x$ and $b=6y$.

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 $84xy$which can be expressed as $2\times 7x\times 6y$.

It follows that middle term is of the form $2ab$.

Step 4. Write the square of the binomial.

So, $49x^2+84xy+36y^2$ 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, $49x^2+84xy+36y^2={\left(7x+6y\right)}^2$.

Step 5. Check by multiplying.

Verification can be obtained as follows.

$\begin{array}{rcl}{\left(7x+6y\right)}^2& =& {\left(7x\right)}^2+2\times 7x\times 6y+{\left(6y\right)}^2\\ & =& 49x^2+84xy+36y^2\end{array}$