Problem 12 Rewrite each expression as a squ... [FREE SOLUTION]

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McGraw Hill

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2004 Edition

Chapter 7: Problem 12

Rewrite each expression as a square with a constant added or subtracted. $$ p^{2}-16 p+60 $$

Short Answer

Expert verified

$ (p-8)^{2} - 4 $

Step by step solution

Identify coefficients

Identify the coefficients of the quadratic expression. Here, for the expression \[ p^{2} - 16p + 60 \] the coefficient of \[ p^{2} \] is 1, the coefficient of \[ p \] is -16, and the constant term is 60.

Complete the square

To complete the square, take the coefficient of \[ p \], which is -16, divide it by 2, and square it. This process gives us: \[ \frac{-16}{2}=-8 \] and \[ (-8)^{2} = 64 \]. Now, rewrite the expression: \[ p^{2} - 16p + 64 - 64 + 60 \].

Group terms to form a perfect square

Group the perfect square trinomial: \[ p^{2} - 16p + 64 \] and combine the constants (-64 + 60): \[ (p-8)^{2} - 4 \].

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

headline of the respective core concept

Quadratic expressions like $ p^{2} - 16p + 60 $ are commonly found in algebra. They are expressions of the form $ ax^{2} + bx + c $. The key components are:

the quadratic term ( $ ax^{2} $ ),
the linear term ( $ bx $ ),
and the constant term ( $ c $ ).

In our example, $ a = 1 $, $ b = -16 $, and $ c = 60 $. Understanding these parts helps us rewrite the expression using completing the square.

headline of the respective core concept

Creating a perfect square trinomial is a powerful way to simplify quadratic expressions. First, focus on the linear term's coefficient ( $ b $ ). In $ p^{2} - 16p + 60 $, $ b = -16 $. To find the term needed to form a perfect square, divide $ b $ by 2, then square it: $ \frac{-16}{2} = -8 $ and $ (-8)^{2} = 64 $. Now, add and subtract 64 within the expression: $ p^{2} - 16p + 64 - 64 + 60 $. This forms $ p^{2} - 16p + 64 $, which is $ (p-8)^{2} $.

headline of the respective core concept

Constant term manipulation is the final step in completing the square. After creating the perfect square trinomial ( $ p^{2} - 16p + 64 = (p-8)^{2} $ ), we need to reconcile the artificial number we added (64). We subtract this added 64 to keep the expression equal to the original. Therefore, the quadratic expression $ p^{2} - 16p + 60 = (p-8)^{2} - 4 $. This method transforms the expression into a perfect square form ( $ (p-8)^{2} $ ) with an adjusted constant (-4).

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91影视

Rewrite each expression as a square with a constant added or subtracted. $$ p^{2}-16 p+60 $$

Short Answer

Step by step solution

Identify coefficients

Complete the square

Group terms to form a perfect square

Key Concepts

headline of the respective core concept

headline of the respective core concept

headline of the respective core concept

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