Problem 74 - Intermediate Algebra

Question

Multiply and then simplify if possible. $$ (\sqrt{3 x+1}+2)^{2} $$

Step-by-Step Solution

Answer

Simplified expression: \(3x + 5 + 4\sqrt{3x+1}\).

Step 1: Understand the Expression

The expression given is \((\sqrt{3x+1} + 2)^2\). We need to multiply this expression by itself and simplify the result if possible.

Step 2: Apply the Binomial Theorem

Use the formula \((a + b)^2 = a^2 + 2ab + b^2\). Here, \(a = \sqrt{3x+1}\) and \(b = 2\). Substitute these values into the formula to expand the expression.

Step 3: Square Each Term

Calculate \(a^2\): \((\sqrt{3x+1})^2 = 3x + 1\). Calculate \(b^2\): \((2)^2 = 4\).

Step 4: Calculate the Middle Term

Find \(2ab\): \(2 \cdot \sqrt{3x+1} \cdot 2 = 4\sqrt{3x+1}\).

Step 5: Combine All Terms

Add all the terms obtained from the previous steps: \(a^2 + 2ab + b^2\). This gives us \(3x + 1 + 4\sqrt{3x+1} + 4\).

Step 6: Simplify the Expression

Combine like terms in the expression: \(3x + 1 + 4\sqrt{3x+1} + 4\) simplifies to \(3x + 5 + 4\sqrt{3x+1}\).

Key Concepts

Squaring Binomials

When you encounter a binomial, which is an expression with two terms, you might need to square it. The process involves using a special formula known as the Binomial Theorem. For example, if you have an expression like \((a + b)^2\), you don't simply square each term; instead, you apply the formula:

• \(a^2\)
• \(2ab\)
• \(b^2\)

This results in \(a^2 + 2ab + b^2\). Each part of this formula represents a component of the expanded binomial. The \(a^2\) is the first term squared, the \(2ab\) is the product of both terms doubled, and the \(b^2\) is the second term squared.
Using this methodology helps prevent common mistakes that happen when mistakenly trying to square each term individually without using the Binomial Theorem. It's an efficient way to expand expressions and is especially useful in algebra when dealing with complex variables.

Algebraic Expressions

An algebraic expression is a mathematical phrase that can include numbers, variables, and operators like addition and multiplication. In the exercise, we have an expression \((\sqrt{3x+1} + 2)^2\) which is a form of algebraic expression.
Algebraic expressions often represent real-world situations and form the building blocks of algebra. Understanding them is crucial as they serve as the foundation for equations and inequalities in mathematics.

• Variables: In this expression, \(x\) is a variable. A variable represents an unknown value that can change.
• Radicals: The square root \(\sqrt{3x+1}\) is a radical expression, showing a value under a root symbol.
• Constants: Numbers like 2 or 4 in the expression are constants, representing known values.

Interpreting these elements correctly makes manipulating and evaluating expressions easier, which is key to solving algebraic problems.

Simplifying Expressions

Simplifying an expression involves reducing it to its most basic form. This means combining like terms and eliminating any unnecessary complexity. The final step of simplifying the exercise expression \(3x + 1 + 4\sqrt{3x+1} + 4\) into \(3x + 5 + 4\sqrt{3x+1}\) shows how this is done.
To simplify an expression, follow these general steps:

• Look for common terms that can be combined, like the constants \(1\) and \(4\) in our example, which simplify to \(5\).
• Simplify any coefficients of terms involving variables or radicals, ensuring that you consolidate all similar parts of your expression.

These steps are vital in algebra because a simplified expression is often easier to work with. It not only looks cleaner but also allows further operations, like solving equations, to be more straightforward. Practice makes perfect, so regularly practicing these steps will enhance your algebra skills profoundly!