Problem 75 - Intermediate Algebra

Question

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

Step-by-Step Solution

Answer

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

Step 1: Expand the Expression

We need to expand the square expression \((\sqrt{2x+5} - 1)^2\). This is similar to expanding \((a-b)^2\), which follows the formula \((a-b)^2 = a^2 - 2ab + b^2\). Here, \(a = \sqrt{2x+5}\) and \(b = 1\).

Step 2: Apply the Squaring Formula

Using the formula from Step 1: \[(\sqrt{2x+5})^2 - 2(\sqrt{2x+5})(1) + 1^2\] This becomes: \[(2x+5) - 2\sqrt{2x+5} + 1\]

Step 3: Simplify the Expression

Combine like terms: - The terms \(5\) and \(1\) are constants, so we can add them together to simplify the expression.- The expression simplifies to:\[2x + 6 - 2\sqrt{2x+5}\] This is the simplest form since \(\sqrt{2x+5}\) cannot be simplified further.

Key Concepts

Square Roots

Square roots are a fundamental concept in mathematics. They allow us to find a number that, when multiplied by itself, equals the original number under the square root symbol. The square root of a number, say \( a \), is denoted as \( \sqrt{a} \). For instance, \( \sqrt{9} = 3 \) because \( 3 \times 3 = 9 \). In algebra, square roots can be applied to expressions involving variables, such as \( \sqrt{2x + 5} \). Here, the expression under the square root, known as the radicand, combines a number (2) and a variable (x). Understanding how to manipulate these expressions is crucial because they appear often in equations and formulas.
When dealing with square roots in algebraic expressions, remember that:

• Square roots can only be simplified if the radicand is a perfect square.
• The square root of a multiplied product can be separated: \( \sqrt{ab} = \sqrt{a} \cdot \sqrt{b} \).
• If possible, re-writing square roots in their simplest form can make solving equations easier.

These concepts are particularly important when simplifying or expanding expressions like the one in our original exercise.

Algebraic Expressions

Algebraic expressions are combinations of numbers, variables, and operations (like addition and multiplication). They are at the heart of algebra, providing a way to describe mathematical relationships. In our exercise, the expression \( \sqrt{2x+5} - 1 \) is an algebraic expression that involves a square root, a subtraction operation, and a variable term.
There are several key components to understand in algebraic expressions:

• **Terms**: Parts of the expression separated by + or - signs. In \( \sqrt{2x+5} - 1 \), \( \sqrt{2x+5} \) and -1 are terms.
• **Coefficients**: Numbers that multiply the variables in an expression. In \( 2x \), 2 is the coefficient.
• **Constant**: A term without variables, like -1 in our expression.
• **Simplification**: Combining like terms (terms with the same variables and powers) to make the expression as simple as possible.

By understanding these parts, students can learn how to manipulate expressions to simplify or expand them, just as needed in our example.

Expansion Formula

The expansion formula is a valuable tool when dealing with expressions that need to be expanded, such as \( (a-b)^2 \). This specific formula is: \[(a-b)^2 = a^2 - 2ab + b^2\] It helps break down the squared expression into individual parts that are easier to work with. In our exercise, \( a = \sqrt{2x+5} \) and \( b = 1 \). Applying the expansion formula here allows us to turn a complex-looking square into a simpler expression. Here's how it works step-by-step:

• **First Term**: Square the first part of the expression: \((\sqrt{2x+5})^2\) results in \(2x+5\).
• **Second Term**: Twice the product of the first and second terms: \(2(\sqrt{2x+5})(1)\).
• **Third Term**: Square the second part: \(b^2 = 1\).

Once expanded, the expression becomes: \((2x+5) - 2\sqrt{2x+5} + 1\). This process not only simplifies the calculation but also illustrates the beauty of algebraic structure, allowing students to untangle what initially appears to be complex.