Mathematical Reasoning

Answer the question in the image with step by step solution.

Here is the step-by-step solution for multiplying and simplifying the given expressions.

Problem 1: Multiply and simplify: (√x + 2√3)²

Solution: We can use the formula for squaring a binomial: (a + b)² = a² + 2ab + b². In this expression, a = √x and b = 2√3.

1. Square the first term (a²): (√x)² = x

2. Find twice the product of the two terms (2ab): 2 * (√x) * (2√3) = 4 * √(x * 3) = 4√(3x)

3. Square the second term (b²): (2√3)² = 2² * (√3)² = 4 * 3 = 12

4. Combine the terms: x + 4√(3x) + 12

The simplified expression is x + 12 + 4√(3x).

Problem 2: Multiply and simplify: (√x + √3)(√x - √3)

Solution: We can use the formula for the difference of squares: (a + b)(a - b) = a² - b². In this expression, a = √x and b = √3.

1. Square the first term (a²): (√x)² = x

2. Square the second term (b²): (√3)² = 3

3. Subtract the second result from the first (a² - b²): x - 3

The simplified expression is x - 3.

Final Answers:

(√x + 2√3)² = x + 12 + 4√(3x)

(√x + √3)(√x - √3) = x - 3