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Ch. R - Review of Basic Concepts
Lial - College Algebra 13th Edition
Lial13th EditionCollege AlgebraISBN: 9780136881063Not the one you use?Change textbook
All textbooksLial 13th EditionCh. R - Review of Basic ConceptsProblem 139
Chapter 1, Problem 139

Find all values of b or c that will make the polynomial a perfect square trinomial. 100r2-60r+c

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Recognize that a perfect square trinomial takes the form \(\left(\sqrt{A}r + d\right)^2 = Ar^2 + 2d\sqrt{A}r + d^2\), where \(A\) is the coefficient of \(r^2\) and \(d\) is a constant to be determined.
Identify the coefficient of \(r^2\) in the given polynomial: \$100r^2\(. So, \)A = 100$ and \(\sqrt{A} = 10\).
Compare the middle term of the polynomial, \(-60r\), to the middle term of the perfect square form, \(2d \times 10r = 20d r\). Set \(20d = -60\) to solve for \(d\).
Solve for \(d\) by dividing both sides of the equation \(20d = -60\) by 20, which gives \(d = -3\).
Find the constant term \(c\) by squaring \(d\): \(c = d^2 = (-3)^2\). This value of \(c\) will make the polynomial a perfect square trinomial.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Perfect Square Trinomial

A perfect square trinomial is a quadratic expression that can be factored into the square of a binomial, typically in the form (ax + b)^2 = a^2x^2 + 2abx + b^2. Recognizing this form helps identify the necessary constant term to complete the square.
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Identifying Coefficients in Quadratics

In a quadratic expression ax^2 + bx + c, the coefficients a, b, and c determine its shape and factorization. Understanding how these coefficients relate, especially the middle term and constant, is essential to determine if the polynomial is a perfect square.
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Completing the Square

Completing the square involves rewriting a quadratic expression so that it forms a perfect square trinomial. This process often requires finding a specific constant term c that makes the expression factorable as (mx + n)^2, which is key to solving the given problem.
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