Problem 58 - Elementary Algebra

Question

Simplify each expression by performing the indicated operation. $$ (3-\sqrt{3 x})^{2} $$

Step-by-Step Solution

Answer

Question: Simplify the expression (3-sqrt(3x))^2. Answer: The simplified expression is 9 - 6√(3x) + 3x.

Step 1: Identify 'a' and 'b' values

In the given expression, \(a = 3\) and \(b = \sqrt{3x}\).

Step 2: Apply the square formula

Using the formula \((a-b)^2 = a^2 - 2ab + b^2\), we plug in the values for \(a\) and \(b\): \((3-\sqrt{3x})^2 = 3^2 - 2(3)(\sqrt{3x}) + (\sqrt{3x})^2\). #Step 2: Simplify the expression#

Step 3: Calculate the square of 'a'

Calculate \(3^2\): \((3-\sqrt{3x})^2 = 9 - 2(3)(\sqrt{3x}) + (\sqrt{3x})^2\).

Step 4: Calculate '2ab'

Calculate \(2(3)(\sqrt{3x})\): \((3-\sqrt{3x})^2 = 9 - 6\sqrt{3x} + (\sqrt{3x})^2\).

Step 5: Calculate the square of 'b'

Calculate \((\sqrt{3x})^2\): \((3-\sqrt{3x})^2 = 9 - 6\sqrt{3x} + 3x\). #Step 3: Combine like terms#

Step 6: Rewrite the simplified expression

The simplified expression is: \((3-\sqrt{3x})^2 = 9 - 6\sqrt{3x} + 3x\).

Key Concepts

Binomial Squares

Understanding binomial squares is crucial when simplifying algebraic expressions that involve squaring two-term expressions, such as \( (a+b)^2 \) or \( (a-b)^2 \). In algebra, a binomial is an expression containing two terms. The square of a binomial is an application of the distributive property and can be written as \( a^2 + 2ab + b^2 \) when dealing with \( (a+b)^2 \) and \( a^2 - 2ab + b^2 \) when dealing with \( (a-b)^2 \) where \(a\) and \(b\) are any numbers or variables.

For example, the exercise \( (3-\sqrt{3x})^2 \) can be approached by identifying \( 3 \) as \(a\) and \( \sqrt{3x} \) as \(b\), then applying the formula to get the squared expression. This method ensures accuracy and assists in simplifying more complex algebraic equations.

Radical Expressions

Radical expressions involve roots, and the most common type is the square root. Simplifying radical expressions is often required in algebra to make the expressions easier to work with. Operations that can be performed on radical expressions include addition, subtraction, multiplication, division, and rationalization.

In the exercise \(3-\sqrt{3x})^{2}\), the term \(\sqrt{3x}\) is a radical expression being squared. Squaring a square root effectively removes the root, simplifying the radical expression to the number beneath it. This is why the \(\sqrt{3x}\) becomes \(3x\) after being squared. Remember that radical expressions follow specific rules for simplification and must be handled carefully to avoid errors.

Algebraic Expressions Simplification

Simplifying algebraic expressions typically involves combining like terms and using arithmetic operations to reduce the expression to its simplest form. Like terms are terms within an expression that have identical variable components and exponents, making them easy to add or subtract.

In our step-by-step solution, we perform algebraic simplification by squaring each term, then using arithmetic to combine the constant terms and separate the radical portions. Simplification helps in evaluating the expressions and solving equations, making the relationship between variables clearer and the overall expression neater.

Square Root Operations

Understanding Squares and Roots

Operations involving square roots can sometimes seem counterintuitive, but they follow precise algebraic rules. When you square a square root, the operations cancel out, and you’re left with the argument (the number inside the square root).

Practical Application in Problem Solving

In the exercise, when we square \(\sqrt{3x}\), the square root and the square cancel each other, resulting in \(3x\), which represents the basic principle of combining square root operations with other algebraic processes. This serves as a foundation for more advanced mathematical topics and practical problem solving in algebra.