Problem 69 - Algebra 2

Question

Multiply. \((5 \sqrt{3}-2)^{2}\)

Step-by-Step Solution

Answer

The result of the expression \((5 \sqrt{3} - 2)^{2}\) is \(79 - 20 \sqrt{3}\).

Step 1: Identify the terms a and b

Looking at the expression \((5 \sqrt{3} -2)^{2}\), it becomes clear that \(a = 5 \sqrt{3}\) and \(b = 2\).

Step 2: Calculate \(a^2\)

We have \(a = 5 \sqrt{3}\), so, \(a^2 = (5 \sqrt{3})^2 = 75\).

Step 3: Calculate \(2ab\)

We have \(a = 5 \sqrt{3}\) and \(b = 2\), so, \(2ab = 2 * 5 \sqrt{3} * 2 = 20 \sqrt{3}\).

Step 4: Calculate \(b^2\)

We have \(b = 2\), so, \(b^2 = 2^2 = 4\).

Step 5: Add up the calculated values

Adding all the calculated values gives the result of the expression: \(75 - 20 \sqrt{3} + 4 = 79 - 20 \sqrt{3}\).

Key Concepts

Understanding Radicals

A radical is simply a way to represent roots of numbers. The most common radical is the square root, symbolized by the radical sign \(\sqrt{}\). In our given problem, one of the terms is \(5 \sqrt{3}\), which means 5 times the square root of 3.
Understanding radicals is crucial because they appear frequently in algebra, particularly when simplifying expressions or solving equations. Here's what you should remember about radicals:

• A square root \(\sqrt{x}\) finds a number that, when multiplied by itself, gives \(x\).
• Radicals can be manipulated and simplified, much like regular numbers. For example, \(\sqrt{4} = 2\) and \(\sqrt{9} = 3\).
• Radicals can be multiplied and divided. If you multiply \(\sqrt{a}\) and \(\sqrt{b}\), you can write \(\sqrt{ab}\).

Learning to work with radicals provides a solid foundation for solving more complex problems involving roots and powers.

Unpacking Binomial Squares

A binomial square is an algebraic expression of the form \((a + b)^2\) or \((a - b)^2\). Squaring a binomial means multiplying it by itself. In this exercise, \((5 \sqrt{3} - 2)^2\) is a binomial square.

When you square a binomial, you use the following pattern:

• The square of the first term: \(a^2\)
• Twice the product of the two terms: \(2ab\)
• The square of the second term: \(b^2\)

Combining these, the final expression becomes \(a^2 - 2ab + b^2\) or \(a^2 + 2ab + b^2\).
Recognizing this pattern helps in calculating the square of any binomial accurately without expanding step by step each time.

Working with Algebraic Expressions

Algebraic expressions are made up of variables, numbers, and operations. In our problem, \((5 \sqrt{3} - 2)^2\) is an algebraic expression involving subtraction and radicals.
Understanding how to manipulate algebraic expressions is key to solving problems in algebra. Here are some important points to remember:

• Identify each term in the expression correctly. Each term in an expression like \(5 \sqrt{3}\) contributes to the final result.
• Break down complex expressions into simpler parts. By solving each part step by step, you reduce the chance of making errors.
• When simplifying, carefully combine like terms and apply the laws of arithmetic correctly.

Mastering these basics makes it easier to tackle even the most challenging algebraic expressions with confidence.