Q 6.45TI. - Intermediate Algebra

Question

Factor: $4x^2+12x+9$.

Step-by-Step Solution

Answer

The Factors of $4x^2+12x+9$ is ${\left(2x+3\right)}^2$, that is $\left(2x+3\right)\left(2x+3\right)$.

Step 1. Given information.

Given an expression $4x^2+12x+9$.

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

First term is $4x^2$ which can be expressed as ${\left(2x\right)}^2$.

Last term is 9 which can be expressed as $3^2$.

Consider $a=2x$ and $b=3$.

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 12x which can be expressed as $2\times 2x\times 3$.

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

Step 4. Write the square of the binomial.

So, $4x^2+12x+9$ 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, $4x^2+12x+9={\left(2x+3\right)}^2$.

Step 5. Check by multiplying.

Verification can be obtained as follows.

$\begin{array}{rcl}{\left(2x+3\right)}^2& =& {\left(2x\right)}^2+2\times 2x\times 3+3^2\\ & =& 4x^2+12x+9\end{array}$