## Intermediate Algebra (12th Edition)

Published by Pearson

# Chapter 5 - Section 5.2 - Factoring Trinomials - 5.2 Exercises: 40

#### Answer

$(5r-9)^2$

#### Work Step by Step

$\bf{\text{Solution Outline:}}$ To factor the quadratic expression $ax^2+bx+c,$ find two numbers whose product is $ac$ and whose sum is $b$. Use these $2$ numbers to decompose the middle term of the quadratic expression and then use factoring by grouping. $\bf{\text{Solution Details:}}$ In the given expression, $25r^2-90r+81 ,$ the value of $ac$ is $25(81)=2025$ and the value of $b$ is $-90 .$ The $2$ numbers that have a product $ac$ and a sum of $b$ are $\{ -45,-45 \}.$ Using these $2$ numbers to decompose the middle term of the given expression results to \begin{array}{l}\require{cancel} 25r^2-45r-45r+81 .\end{array} Grouping the first and third terms and the second and fourth terms, the given expression is equivalent to \begin{array}{l}\require{cancel} (25r^2-45r)-(45r-81) .\end{array} Factoring the $GCF$ in each group results to \begin{array}{l}\require{cancel} 5r(5r-9)-9(5r-9) .\end{array} Factoring the $GCF= (5r-9)$ of the entire expression above results to \begin{array}{l}\require{cancel} (5r-9)(5r-9) \\\\= (5r-9)^2 .\end{array}

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