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Chapter 8: Problem 76

Simplify. $$(\sqrt{5 x}+1)(\sqrt{5 x}+1)$$

Short Answer

Expert verified
5x + 2\sqrt{5x} + 1

Step by step solution

01

- Recognize the Form

Identify that \( \sqrt{5x} + 1 \) is being multiplied by itself. This is a binomial squared, so it can be written as \( (a + b)^2 \), where \( a = \sqrt{5x} \) and \( b = 1 \).
02

- Expand the Expression

Apply the formula for the square of a binomial: \[ (a + b)^2 = a^2 + 2ab + b^2 \]. Substitute \( a = \sqrt{5x} \) and \( b = 1 \) into the formula: \[ (\sqrt{5x})^2 + 2(\sqrt{5x})(1) + 1^2 \]
03

- Simplify Each Term

Simplify \( (\sqrt{5x})^2 \) to \( 5x \). Simplify \( 2(\sqrt{5x})(1) \) to \( 2\sqrt{5x} \). Simplify \( 1^2 \) to \( 1 \). The expression now is \[ 5x + 2\sqrt{5x} + 1 \]

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Key Concepts

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

Simplifying Expressions
Simplification helps make algebraic expressions more manageable. Begin by recognizing familiar patterns. Here, our goal is to simplify the binomial squared. According to the formula \((a + b)^2 = a^2 + 2ab + b^2\), each term must be simplified individually.
1. Square \(a = \sqrt{5x}\): \( (\sqrt{5x})^2 = 5x\).
2. Multiply twice with both terms: \(2 \times \sqrt{5x} \times 1 = 2\root(5x)\).
3. Square \(b = 1\), yielding \(1^2 = 1\). Assembling these gives us the simplified form: \5x + 2\root5x + 1\.
Each step breaks the problem into smaller, easier pieces, taking advantage of known formulas.
Expanding Expressions
Expanding an expression involves distributing and combining like terms. Applying formulas helps make complex expressions easier. For example, when given \( (a + b)(a + b) \), expand using algebra rules:
1. Multiply each part (distributing): \( (a + b)(a + b) = a(a + b) + b(a + b)\).
2. Perform multiplications: \((a + b) + (b + b)\).
When expanded fully, the original exercise becomes \( (\root5x)2)(1) + \root5x (1^2\). Combining and simplifying each part results in a cleaner, more understandable expression. Expanding expressions makes solving more straightforward.

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