Explanation Given Information: The equation is r 4 = 2 r 3 + 15 r 2 . 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.” Calculation: Consider the equation is, r 4 = 2 r 3 + 15 r 2 Note that, the equation is not in the standard form. Convert the equation into standard form. Subtract 2 r 3 and 15 r 2 on both sides and perform the operations, r 4 − 2 r 3 − 15 r 2 = 2 r 3 + 15 r 2 − 2 r 3 − 15 r 2 r 4 − 2 r 3 − 15 r 2 = 0 Factor out the greatest common factor r 2 , r 2 ( r 2 − 2 r − 15 ) = 0 Factor the trinomial r 2 − 2 r − 15 . Find the two integers whose product is ( 1 ) ⋅ ( − 15 ) = − 15 and whose sum is −2, The required integers are −5 and 3. Write the middle term by using these terms as, − 2 r = − 5 r + 3 r So, r 2 ( r 2 − 5 r + 3 r − 15 ) = 0 Group the first two terms and the last two terms, r 2 [ ( r 2 − 5 r ) + ( 3 r − 15 ) ] = 0 Factor each group to solve further, r 2 [ r ( r − 5 ) + 3 ( r − 5 ) ] = 0 Factor out ( r − 5 ) . r 2 ( r − 5 ) ( r + 3 ) = 0 Here, the product r 2 ( r − 5 ) ( r + 3 ) is equal to zero. Apply the definition of extension of zero-factor property, The product of the factors r 2 , ( r − 5 ) and ( r + 3 ) is 0 only if at least one of the factors must be 0. That is, r 2 = 0 , r − 5 = 0 or r + 3 = 0 . Solve the equation, r 2 = 0 Again, apply the definition of zero-factor property, r 2 = 0 r ⋅ r = 0 Then, r = 0 or r = 0 That is, r = 0 is the double solution of the equation

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 71E

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Chapter 5.5, Problem 70E

Chapter 5.5, Problem 72E

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The equation r 4 = 2 r 3 + 15 r 2 and check the solutions.

Solution Summary: The author explains the steps for solving the quadratic equation by zero factor property: Convert the equation into standard form, complete the factoring, and write the solution set.

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To calculate: The equation r4=2r3+15r2 and check the solutions.

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