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Chapter 3: Problem 16

Expand the indicated expression. $$ (5+\sqrt{x})^{2} $$

Short Answer

Expert verified
The short answer based on the given step-by-step solution is: Expanding the expression \((5+\sqrt{x})^2\) using the formula for squaring a binomial, we get: \((5+\sqrt{x})^2 = 25 + 10\sqrt{x} + x\).

Step by step solution

01

Identify the values of a and b

In order to apply the formula for squaring a binomial, we must first identify the values of a and b in the given expression: \[ (5+\sqrt{x})^2 \] Here, \(a = 5\) and \(b = \sqrt{x}\).
02

Apply the formula for squaring a binomial

Now that we have identified the values of a and b, we can apply the formula for squaring a binomial: \[ (a+b)^2 = a^2 + 2ab + b^2 \] Substitute the values of a and b from step 1: \[ (5+\sqrt{x})^2 = 5^2 + 2(5)(\sqrt{x}) + (\sqrt{x})^2 \]
03

Simplify the expression

Next, we will simplify the expression to get the final expanded form: \[ (5+\sqrt{x})^2 = 25 + 10\sqrt{x} + x \] The expanded expression is: \[ (5+\sqrt{x})^2 = 25 + 10\sqrt{x} + x \]

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Key Concepts

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

Squaring a Binomial
Squaring a binomial involves taking a binomial expression and raising it to the power of two. A binomial is an algebraic expression containing two terms, such as \((5 + \sqrt{x})\). To square a binomial like this, we follow a specific formula. This formula helps us break down the expression into easier parts so we can manage them easily. It goes like this:
  • Identify the terms: Let \(a\) be the first term, here it's \(5\), and \(b\) be the second term, which is \(\sqrt{x}\).
  • Apply the squaring formula: \((a+b)^2 = a^2 + 2ab + b^2\).
This formula expands our binomial by considering not just each term squared, but also their interaction term \(2ab\), ensuring nothing is missed in the expansion process. This method is universally applicable for any binomial.
Algebraic Expression
An algebraic expression is a combination of numbers, variables, and arithmetic operations. It represents a mathematical phrase that can be simplified or evaluated. In our problem, the expression \((5 + \sqrt{x})\) consists of two distinct elements:
  • \(5\), a constant that remains the same no matter what.
  • \(\sqrt{x}\), which is a variable term indicating the square root of some number \(x\).
Algebraic expressions can represent real-world scenarios or abstract math problems. They are not tied to equality or inequality but can be manipulated to solve equations or to expand like we are doing here with binomial squaring. Understanding how these expressions work is essential as it gives us the foundation to deepen into layers like polynomials, factoring, and higher-degree equations.
Simplifying Expressions
Simplifying expressions involves making them as neat and straightforward as possible. After applying a formula like squaring a binomial, simplification helps to tidy up the resulting expression. The aim is to condense it into an easily interpretable form. Our example \((5+\sqrt{x})^2 = 25 + 10\sqrt{x} + x\) showcases this clearly:
  • First, square the constant \(5\) resulting in \(25\).
  • Then compute the middle term: \(2 \cdot 5 \cdot \sqrt{x} = 10\sqrt{x}\).
  • Finally, square the variable portion \((\sqrt{x})^2=x\).
Each of these steps breaks down complex parts into simpler, solvable bits. By simplifying, we transform a potentially complex expression into a more digestible and familiar form which can be vital for solving equations or for further mathematical operations.

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