Perform the indicated operations. Assume all variables represent positive real numbers. (√2 + 3) (√2 - 3)

Verified step by step guidance

1. Recognize that the expression \((\sqrt{2} + 3)(\sqrt{2} - 3)\) is in the form of a product of conjugates, which follows the pattern \((a + b)(a - b) = a^2 - b^2\).
2. Identify \(a = \sqrt{2}\) and \(b = 3\) from the given expression.
3. Apply the difference of squares formula: \( (\sqrt{2})^2 - (3)^2 \).
4. Calculate each square separately: \((\sqrt{2})^2 = 2\) and \(3^2 = 9\).
5. Subtract the squares: \(2 - 9\) to simplify the expression further.

Key Concepts

Product of Binomials

Multiplying two binomials involves applying the distributive property (FOIL method), where each term in the first binomial is multiplied by each term in the second. This process helps expand expressions like (a + b)(c + d) into a simplified polynomial.

Difference of Squares

The expression (√2 + 3)(√2 - 3) fits the difference of squares pattern: (x + y)(x - y) = x² - y². Recognizing this pattern allows quick simplification by subtracting the square of the second term from the square of the first.

Simplifying Radicals

Simplifying radicals involves evaluating or rewriting square roots in simplest form. For example, (√2)² equals 2, which is essential when applying the difference of squares formula to expressions containing radicals.