Intermediate Algebra (12th Edition)

Published by Pearson
ISBN 10: 0321969359
ISBN 13: 978-0-32196-935-4

Chapter 5 - Section 5.5 - Solving Equations by the Zero-Factor Property - 5.5 Exercises - Page 355: 17


$x=\left\{ 0,6 \right\}$

Work Step by Step

$\bf{\text{Solution Outline:}}$ Express the given equation, $ 6x^2-36x=0 ,$ in factored form. Then, use the Zero-Factor Property by equating each factor to zero. Finally, solve each equation. $\bf{\text{Solution Details:}}$ The factored form of the equation above is \begin{array}{l}\require{cancel} 6x(x-6)=0 .\end{array} Equating each factor to zero (Zero-Factor Property), then \begin{array}{l}\require{cancel} 6x=0 \text{ OR } x-6=0 .\end{array} Using the properties of equality to solve each of the equation above results to \begin{array}{l}\require{cancel} 6x=0 \\\\ x=\dfrac{0}{6} \\\\ x=0 \\\\\text{ OR }\\\\ x-6=0 \\\\ x=6 .\end{array} Hence, the solutions are $ x=\left\{ 0,6 \right\} .$ $\bf{\text{Supplementary Solution/s:}}$ In the expression $ 6x^2-36x ,$ the $GCF$ of the constants of the terms $\{ 6, -36 \}$ is $ 6 .$ The $GCF$ of the common variable/s is the variable/s with the lowest exponent. Hence, the $GCF$ of the common variable/s $\{ x^2,x \}$ is $ x .$ Hence, the entire expression has $GCF= 6x .$ Factoring the $GCF= 6x ,$ the expression above is equivalent to \begin{array}{l}\require{cancel} 6x \left( \dfrac{6x^2}{6x}-\dfrac{36x}{6x} \right) .\end{array} Using the Quotient Rule of the laws of exponents which states that $\dfrac{x^m}{x^n}=x^{m-n},$ the expression above simplifies to \begin{array}{l}\require{cancel} 6x \left( x^{2-1}-6x^{1-1} \right) \\\\= 6x \left( x^{1}-6x^{0} \right) \\\\= 6x \left( x-6(1) \right) \\\\= 6x \left( x-6 \right) .\end{array}
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