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Chapter 10: Problem 75

Multiply, and then simplify if possible. \((\sqrt{2 x+5}-1)^{2}\)

Short Answer

Expert verified
The simplified expression is \(2x + 6 - 2\sqrt{2x + 5}\).

Step by step solution

01

Expand the Expression

To multiply, first recognize that we have a binomial square. Use the formula \[(a-b)^2 = a^2 - 2ab + b^2\] where \(a = \sqrt{2x + 5}\) and \(b = 1\). Thus, expand the expression to:\[(\sqrt{2x + 5})^2 - 2 \times \sqrt{2x + 5} \times 1 + 1^2\].
02

Simplify Each Term

Now let's calculate and simplify each term:- \((\sqrt{2x + 5})^2\) becomes \(2x + 5\) because squaring a square root cancels the square root.- \(-2 \times \sqrt{2x + 5} \times 1\) simplifies to \(-2\sqrt{2x + 5}\).- \(1^2\) simplifies to \(1\).
03

Combine the Terms

Now, combine all the terms obtained from the previous step:\[2x + 5 - 2\sqrt{2x + 5} + 1\].
04

Simplify the Expression

Combine like terms, which in this case are the constant terms:\[2x + 6 - 2\sqrt{2x + 5}\]. This is the final simplified expression for the given problem.

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Key Concepts

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

Simplifying Expressions
Simplifying expressions is a crucial part of algebra that helps in reducing complex expressions into a more manageable form. In our original exercise, simplifying the expression involves both the combination and elimination of terms.
To effectively simplify an expression, follow these steps:
  • Identify and group like terms. Like terms have the same variables raised to the same power. In our case, the terms \(5\) and \(1\) are constants and can be combined.
  • Apply algebraic operations to reduce the expression further. For example, when we simplify \(5 + 1\), we get \(6\).
  • Ensure all terms are expressed in their simplest form. For instance, the term \(2\sqrt{2x + 5}\) is left as is because it cannot be simplified further.
Simplifying helps not just in making calculations easier but also in gaining a clearer insight into the algebraic relationships at play. This is key to mastering more advanced topics.
Algebraic Expansion
Algebraic expansion relates to multiplying out expressions, especially when dealing with terms that are raised to a power, such as binomials. In this exercise, we multiplied a binomial using the formula: \[(a-b)^2 = a^2 - 2ab + b^2\]Expanding a binomial squared means turning it into a sum or difference of terms. Here's a recap of how this works in the example:
  • The expression \((\sqrt{2x+5} - 1)^2\) is expanded by substituting \(a = \sqrt{2x+5}\) and \(b = 1\).
  • Each part of the formula is calculated separately: \(a^2 = (\sqrt{2x+5})^2\), which simplifies to \(2x+5\), \(-2ab = -2 \times \sqrt{2x+5} \times 1\) becomes \(-2\sqrt{2x+5}\), and \(b^2 = 1^2\) which simplifies to \(1\).
  • Finally, these simplified terms are combined to form the expanded expression \(2x + 5 - 2\sqrt{2x + 5} + 1\).
The ability to expand binomials is essential in algebra as it lays the groundwork for solving equations and understanding polynomial expressions.
Square Roots
Square roots are fundamental in simplifying mathematical expressions, especially when working with quadratic equations and polynomial functions. A square root of a number \(a\), written as \(\sqrt{a}\), is a value that, when multiplied by itself, gives \(a\). In this particular exercise, square roots play a significant role:
  • The expression starts with a square root: \(\sqrt{2x+5}\).
  • When squaring \(\sqrt{2x + 5}\), it simplifies directly to \(2x + 5\) because the operation of squaring and taking the square root cancels each other out.
  • Understanding how square roots interact with other operations, such as multiplication and addition, is key. In this exercise, the expression \(2\sqrt{2x + 5}\) is handled as an intact term, showing that while it cannot be simplified further algebraically, it is crucial to the overall expression.
This knowledge is not only applicable here but also extends to solving real-world problems where roots and squares are involved.

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Most popular questions from this chapter

Use rational exponents to simplify each radical. Assume that all variables represent positive numbers. $$ \sqrt[10]{a^{5} b^{5}} $$

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Factor the common factor from the given expression. $$ x^{1 / 5} ; x^{2 / 5}-3 x^{1 / 5} $$

Identify the domain and then graph each function. $$f(x)=\sqrt[3]{x}-2$$

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