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Intermediate Algebra

OPENSTAX

$Math Studyset 91影视 Explanations$ Math

OER 2017 Edition 路 1346 pages

ISBN: 9780998625720

Chapter 6: Q 6.45TI. (page 593)

Factor: $4 x^{2} + 12 x + 9$ .

Short Answer

Expert verified

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

Step by step solution

01

Step 1. Given information.

Given an expression $4 x^{2} + 12 x + 9$ .

02

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

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

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

Consider $a = 2 x$ and $b = 3$ .

So, first and last term are perfect squares.

03

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

Here, middle term is 12x which can be expressed as $2 脳 2 x 脳 3$ .

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

04

Step 4. Write the square of the binomial.

So, $4 x^{2} + 12 x + 9$ is a perfect square trinomial of the form $a^{2} + 2 a b + b^{2}$ .

Note that ${(a + b)}^{2} = a^{2} + 2 a b + b^{2}$ .

Therefore, $4 x^{2} + 12 x + 9 = {(2 x + 3)}^{2}$ .

05

Step 5. Check by multiplying.

Verification can be obtained as follows.

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

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