Multiply. See Example 7.(√5 + 2)²

Verified step by step guidance

1. Recognize that the expression \((\sqrt{5} + 2)^2\) is a binomial squared, which can be expanded using the formula \((a + b)^2 = a^2 + 2ab + b^2\).
2. Identify \(a\) as \(\sqrt{5}\) and \(b\) as \(2\) in the expression \((\sqrt{5} + 2)^2\).
3. Calculate \(a^2\) which is \((\sqrt{5})^2 = 5\).
4. Calculate \(2ab\) which is \(2 \times \sqrt{5} \times 2 = 4\sqrt{5}\).
5. Calculate \(b^2\) which is \(2^2 = 4\).

Key Concepts

Binomial Expansion

Binomial expansion refers to the process of expanding expressions that are raised to a power, particularly those in the form of (a + b)². The formula for this expansion is (a + b)² = a² + 2ab + b², which allows us to calculate the square of a binomial by squaring each term and adding twice the product of the two terms.

Square Roots

Square roots are a fundamental concept in mathematics, representing a value that, when multiplied by itself, gives the original number. In the expression √5, the square root of 5 is an irrational number approximately equal to 2.236. Understanding square roots is essential for simplifying expressions that involve radical terms.

Algebraic Simplification

Algebraic simplification involves reducing expressions to their simplest form by combining like terms and applying arithmetic operations. In the context of the given expression, simplifying (√5 + 2)² requires careful application of the binomial expansion and combining the resulting terms to achieve a concise final expression.