https://file.aiquickdraw.com/mathbot/file/1729182167498250t0ho3.jpg

Let's work through both parts of the problem step by step:

### Part 1:
\[
(\sqrt{x} - 2\sqrt{5})(\sqrt{x} + 2\sqrt{5})
\]

This is a difference of squares, which has the general form:
\[
(a - b)(a + b) = a^2 - b^2
\]
In this case:
- \(a = \sqrt{x}\)
- \(b = 2\sqrt{5}\)

So, applying the difference of squares formula:
\[
(\sqrt{x})^2 - (2\sqrt{5})^2 = x - 4 \times 5 = x - 20
\]
Thus, the result of the first expression is:
\[
x - 20
\]

### Part 2:
\[
(\sqrt{x} + \sqrt{5})^2
\]
This is a binomial square, which follows the formula:
\[
(a + b)^2 = a^2 + 2ab + b^2
\]
In this case:
- \(a = \sqrt{x}\)
- \(b = \sqrt{5}\)

Expanding it:
\[
(\sqrt{x})^2 + 2(\sqrt{x})(\sqrt{5}) + (\sqrt{5})^2
\]
Simplifying each term:
\[
x + 2\sqrt{5x} + 5
\]
Thus, the result of the second expression is:
\[
x + 2\sqrt{5x} + 5
\]

### Final Answers:
1. \((\sqrt{x} - 2\sqrt{5})(\sqrt{x} + 2\sqrt{5}) = x - 20\)
2. \((\sqrt{x} + \sqrt{5})^2 = x + 2\sqrt{5x} + 5\)

Would you like further clarification or details?

### Relative questions:
1. How would you expand \((\sqrt{x} + 3)(\sqrt{x} - 3)\)?
2. What is the general form for \((a - b)^2\)?
3. How do you apply the distributive property to simplify binomials?
4. What happens when you multiply two terms involving square roots like \(\sqrt{a} \cdot \sqrt{b}\)?
5. How does the difference of squares formula simplify certain quadratic expressions?

### Tip:
Remember, when multiplying square roots, you can combine them under a single square root: \(\sqrt{a} \cdot \sqrt{b} = \sqrt{ab}\).

(√x - 2√5)(√x + 2√5) and (√x + √5)^2

Learn how to simplify expressions like (√x - 2√5)(√x + 2√5) and (√x + √5)^2 using the difference of squares and binomial expansion. This step-by-step solution covers algebraic manipulation of radicals. Suitable for students in Grades 10-12.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation