Problem 57 - Elementary Algebra

Question

Simplify each expression by performing the indicated operation. $$ (2+\sqrt{5 x})^{2} $$

Step-by-Step Solution

Answer

Question: Simplify the expression \((2+\sqrt{5x})^{2}\). Answer: The simplified expression is \(4 + 4\sqrt{5x} + 5x\).

Step 1: Apply the binomial expansion formula

The binomial expansion formula for the square of a binomial expression \((a+b)^2\) is given by \((a+b)^2 = a^2 + 2ab + b^2\). Let's apply this formula to our expression, where \(a=2\) and \(b=\sqrt{5x}\).

Step 2: Square the first term

Square the first term (2) to get \(2^2 = 4\).

Step 3: Square the second term

Square the second term (\(\sqrt{5x}\)) to get \((\sqrt{5x})^2 = 5x\).

Step 4: Multiply the terms

Multiply the two terms (2 and \(\sqrt{5x}\)) to get \(2\cdot\sqrt{5x} = 2\sqrt{5x}\). Since we have 2 instances of this multiplication in the formula (\(2ab\)), we need to double the result: \(2\cdot2\sqrt{5x} = 4\sqrt{5x}\).

Step 5: Add the results

Now we can combine the results from steps 2, 3, and 4 to get the simplified expression: \((2+\sqrt{5x})^2 = 4 + 4\sqrt{5x} + 5x\).

Key Concepts

Algebraic Expressions

Algebraic expressions are a fundamental part of mathematics with diverse components, including variables, constants, and operators like addition, subtraction, multiplication, and division. They are formed by combining these elements to represent mathematical concepts and relationships. In the exercise we are dealing with, the expression \((2+\sqrt{5x})^2\) is an example of a binomial expression. Algebraic expressions can vary in complexity, from simple monomials like \(3x\) to complex combinations involving exponents and radicals. Understanding how to interpret each part of an expression helps in performing further simplifications and operations such as expansion, factorization, and evaluation. Recognizing patterns like the square of a binomial expression, as we've seen in the exercise, allows us to quickly apply the correct binomial formula. This formula helps to break down and manage expressions systematically, laying the groundwork for effective simplification.

Simplification

Simplification is an essential technique in algebra that makes expressions easier to work with by reducing them to their simplest form. It involves combining like terms, applying arithmetic operations, and sometimes factoring.In our exercise, simplification begins by recognizing the pattern \((a+b)^2\) and then applying the binomial expansion formula. We must carefully perform each step:

• Square the individual terms separately.
• Apply arithmetic operations on the coefficients and variables.
• Add or subtract the resulting terms to finalize the simplified expression.

As exemplified in our work, applying these methods allows us to transform \((2+\sqrt{5x})^2\) into its simplest form: \(4 + 4\sqrt{5x} + 5x\). Knowing how to simplify expressions can lead to more effortless problem-solving and a deeper understanding of algebraic relationships.

Mathematical Operations

Mathematical operations are critical actions we perform on numbers or algebraic expressions to solve problems or to simplify expressions further. In algebra, these operations typically include addition, subtraction, multiplication, and division.When solving expressions like the one in our example, we interchangeably use these operations at each step:

• Multiplication: Used when expanding the binomial using \((a+b)^2 = a^2 + 2ab + b^2\).
• Addition: Applied when summing the terms obtained after individual operations to reach the final simplified form.

For instance, during expansion, the multiplication and subsequent addition lead to the combination of terms \(4 + 4\sqrt{5x} + 5x\). Mastering these mathematical operations is essential, as they allow you to handle complex expressions efficiently and pave the way for successful algebraic manipulations.