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Chapter 10: Problem 22

How are the problems Multiply \((y-5)^{2}\) and Multiply \((\sqrt{7}-2)^{2}\) similar? What method can be used to multiply each of them?

Short Answer

Expert verified
Both expressions \((y-5)^2\) and \((\sqrt{7}-2)^2\) are similar because they are in the form of \((a-b)^2\). The square of binomials can be expanded using the identity \((a-b)^2 = a^2 - 2ab + b^2\). After expanding both expressions, we can observe that they both have a middle term with a coefficient of -2.

Step by step solution

01

Identify the expressions

We have two expressions: 1. \((y-5)^2\) 2. \((\sqrt{7}-2)^2\) Both expressions are in the form of \((a-b)^2\).
02

Identify "a" and "b" in both expressions

In the first expression \((y-5)^2\): a = y, b = 5 In the second expression \((\sqrt{7}-2)^2\): a = \(\sqrt{7}\), b = 2
03

Apply the identity \((a-b)^2 = a^2 - 2ab + b^2\)

Expand and simplify the first expression: \((y-5)^2 = y^2 - 2(y)(5) + 5^2\) \((y-5)^2 = y^2 - 10y + 25\) Expand and simplify the second expression: \((\sqrt{7}-2)^2 = (\sqrt{7})^2 - 2(\sqrt{7})(2) + 2^2\) \((\sqrt{7}-2)^2 = 7 - 4\sqrt{7} + 4\)
04

Explain the similarities between the expanded expressions

Both expressions are in the form of \(a^2 - 2ab + b^2\). In both expressions, the coefficient of the middle term is -2. For the first expression: Expanded form: \(y^2 - 10y + 25\) Middle term: \(-10y\) For the second expression: Expanded form: \(7 - 4\sqrt{7} + 4\) Middle term: \(-4\sqrt{7}\)
05

Summarize the findings

Both expressions \((y-5)^2\) and \((\sqrt{7}-2)^2\) are similar because they are in the form of \((a-b)^2\) and can be expanded using the identity \((a-b)^2 = a^2 - 2ab + b^2\). After expanding both expressions, we can observe that they both have a middle term with a coefficient of -2.

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Most popular questions from this chapter

Rationalize the denominator and simplify completely. Assume the variables represent positive real numbers. $$\frac{b-25}{\sqrt{b}-5}$$

Rationalize the denominator of each expression. Assume all variables represent positive real numbers. $$\frac{9}{\sqrt[3]{25}}$$

Rationalize the denominator of each expression. Assume all variables represent positive real numbers. $$\frac{21}{\sqrt[3]{3}}$$

Multiply and simplify. $$\sqrt{\frac{6}{5}} \cdot \sqrt{\frac{1}{8}}$$

Rationalize the denominator of each expression. $$\frac{25}{\sqrt{10}}$$
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