Explanation Given Information: The equation is ( x + 4 ) ( x + 7 ) = 10 . Formula used: “There are following steps for solving the quadratic equation by zero factor property: Step1: Convert the equation into standard form. Step 2: Completely factor the equation. Step 3: Use the zero-factor property to established each factor with a variable equal to 0. Step 4: Solve the resultant equations. Step 5: Check each result in the unique equation. Write the solution set.” Extension of Zero-Factor Property: “If the product of more than two numbers is zero, then at least one of the numbers must be zero.” Distributive Property: a ( b + c ) = a b + b c . Calculation: Consider the equation is, ( x + 4 ) ( x + 7 ) = 10 Note that, the equation is not in the standard form. Convert the equation into standard form. Apply distributive property on left side and simplify the terms, x ( x + 7 ) + 4 ( x + 7 ) = 10 x 2 + 7 x + 4 x + 28 = 10 x 2 + 11 x + 28 = 10 Subtract 10 onboth sides and perform the operations, x 2 + 11 x + 28 − 10 = 10 − 10 x 2 + 11 x + 18 = 0 Find the two integers whose product is ( 1 ) ⋅ ( 18 ) = 18 and whose sum is 11, The required integers are 2 and 9. Write the middle term by using these terms as, 11 x = 2 x + 9 x So, x 2 + 2 x + 9 x + 18 = 0 Group the first two terms and the last two terms, ( x 2 + 2 x ) + ( 9 x + 18 ) = 0 Factor each group to solve further, x ( x + 2 ) + 9 ( x + 2 ) = 0 Factor out ( x + 2 ) , ( x + 2 ) ( x + 9 ) = 0 Here, the product ( x + 2 ) ( x + 9 ) is equal to zero. Apply the definition of zero-factor property, The product of the factors ( x + 2 ) and ( x + 9 ) is 0 only if at least one of the factors must be 0

Beginning and Intermediate Algebra (6th Edition)

6th Edition

ISBN: 9780321969163

Author: Margaret L. Lial, John Hornsby, Terry McGinnis

Publisher: PEARSON

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Chapter 5.5, Problem 61E

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Chapter 5.5, Problem 60E

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The equation ( x + 4 ) ( x + 7 ) = 10 and check the solutions.

Solution Summary: The author explains the steps for solving the quadratic equation by zero-factor property.

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