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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. 159 (page 603)

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

Short Answer

Expert verified

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

Step by step solution

01

Step 1. Given information.

Given an expression $16 y^{2} + 24 y + 9$ .

02

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

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

Consider $a = 4 y$ and $b = 3$ .

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

Therefore, first and last terms are perfect squares.

03

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

Middle term of the expression is $24 y$ .

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

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

04

Step 4. Write the square of the binomial.

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

Obtain the result as follows.

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

05

Step 5. Check the result.

Verify the result as follows.

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

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