Q. 159 - Intermediate Algebra

Question

Factor $16y^2+24y+9$ completely using the perfect square trinomials pattern.

Step-by-Step Solution

Answer

Factors of $16y^2+24y+9$ are ${\left(4y+3\right)}^2$.

Step 1. Given information.

Given an expression $16y^2+24y+9$.

Step 2. Check if the first and last term of the expression a perfect square.

First term and last term are $16y^2$ and 9 respectively.

Consider $a=4y$ and $b=3$.

It can be seen that $16y^2={\left(4y\right)}^2$ and $9=3^2$.

Therefore, first and last terms are perfect squares.

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

Middle term of the expression is $24y$.

Also, $24y=2\left(4y\right)\left(3\right)$.

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

Step 4. Write the square of the binomial.

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

Obtain the result as follows.

$\begin{array}{rcl}16y^2+24y+9& =& {\left(4y\right)}^2+2\left(4y\right)\left(3\right)+3^2\\ & =& {\left(4y+3\right)}^2\end{array}$

Step 5. Check the result.

Verify the result as follows.

$\begin{array}{rcl}{\left(4y+3\right)}^2& =& {\left(4y\right)}^2+2\left(4y\right)3+3^2\\ & =& 16y^2+24y+9\end{array}$