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Chapter 3: Problem 18

Find each product. $$[x-(3-\sqrt{5})][x-(3+\sqrt{5})]$$

Short Answer

Expert verified
The product is \[x^2 - 6x + 4\].

Step by step solution

01

Recognize the Conjugates

Notice that the two binomials are conjugates: \[a+b\] and \[a-b\]. Here, \[a=3\] and \[b= \sqrt{5}\].
02

Apply the Product of Conjugates Formula

The product of two conjugates is given by the formula \[(a+b)(a-b) = a^2 - b^2\].
03

Substitute the Values

Substitute \[a = 3\] and \[b = \sqrt{5}\] into the formula: \[a^2 - b^2 = 3^2 - (\sqrt{5})^2\].
04

Simplify the Expression

Calculate \[3^2 = 9\] and \[(\sqrt{5})^2 = 5\]. Therefore, \[9 - 5 = 4\].

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Key Concepts

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

Binomials
Binomials are algebraic expressions containing two terms. These terms are often separated by a plus (+) or minus (-) sign. Understanding binomials is crucial for various algebraic operations. For example, consider \(x-(3-\sqrt{5})\) and \(x-(3+\sqrt{5})\). Here, each expression has two terms, making them binomials. Binomials are commonly encountered in polynomial expressions and are fundamental in algebraic manipulations.
Square Roots
Square roots are mathematical symbols that represent a value which, when multiplied by itself, gives the original number. The square root of a number \(n\) is denoted as \(\sqrt{n}\). For example, \(\sqrt{5}\) is a square root because \(\sqrt{5} \cdot \sqrt{5} = 5\). Square roots play an essential role in algebra and particularly in problems involving binomials. In our example, the term \(3-\sqrt{5}\) includes a square root, making the expression more complex but interesting to simplify.
Conjugates
Conjugates are pairs of expressions that have the same terms but opposite signs between them. They often appear in the form \(a+b\) and \(a-b\). These pairs are especially useful in simplifying expressions and rationalizing denominators. For instance, in the problem \(x-(3-\sqrt{5})\) and \(x-(3+\sqrt{5})\) are conjugates. Recognizing conjugates allows you to use specific algebraic formulas to simplify expressions easily.
Simplification
Simplification is the process of reducing an algebraic expression into its simplest form. This often involves combining like terms, factoring, and using algebraic identities. In solving the given problem, recognizing that the binomials are conjugates helps simplify the expression using the formula \((a+b)(a-b)=a^{2} - b^{2}\). Substituting \(a = 3\) and \(b = \sqrt{5}\) into this formula: \((3+\sqrt{5})(3-\sqrt{5}) = 3^{2} - (\sqrt{5})^{2}\), we get \{9 - 5 = 4\}. This simplification makes complex expressions more manageable and easier to solve.

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