https://r2.erweima.ai/mathbot/image/1726503534053m291dzfs.jpg

Let's go step by step through each part of the image and complete the required tables.

### Table 1: Expanding squares of binomials \( (a + b)^2 \)
The formula to use is:

\[
(a + b)^2 = a^2 + 2ab + b^2
\]

| Expression      | \(a\)  | \(b\)  | \(a^2\)   | \(b^2\)   | \(2ab\)  | \(a^2 + 2ab + b^2\) |
|-----------------|--------|--------|-----------|-----------|----------|---------------------|
| \( (3x + 2)^2 \) | \(3x\) | \(2\)  | \(9x^2\)  | \(4\)     | \(12x\)  | \(9x^2 + 12x + 4\)  |
| \( (4x + 1)^2 \) | \(4x\) | \(1\)  | \(16x^2\) | \(1\)     | \(8x\)   | \(16x^2 + 8x + 1\)  |
| \( (5x + 1)^2 \) | \(5x\) | \(1\)  | \(25x^2\) | \(1\)     | \(10x\)  | \(25x^2 + 10x + 1\) |

### Table 2: Expanding squares of binomials \( (a - b)^2 \)
The formula to use here is:

\[
(a - b)^2 = a^2 - 2ab + b^2
\]

| Expression      | \(a\)  | \(b\)  | \(a^2\)   | \(b^2\)   | \(2ab\)  | \(a^2 - 2ab + b^2\) |
|-----------------|--------|--------|-----------|-----------|----------|---------------------|
| \( (x - 5)^2 \)  | \(x\)  | \(5\)  | \(x^2\)   | \(25\)    | \(10x\)  | \(x^2 - 10x + 25\)  |
| \( (2x - 4)^2 \) | \(2x\) | \(4\)  | \(4x^2\)  | \(16\)    | \(16x\)  | \(4x^2 - 16x + 16\) |
| \( (4x - 3)^2 \) | \(4x\) | \(3\)  | \(16x^2\) | \(9\)     | \(24x\)  | \(16x^2 - 24x + 9\) |

### Table 3: Product of binomials \( (a + b)(a - b) \)
Here, the formula is:

\[
(a + b)(a - b) = a^2 - b^2
\]

| Expression             | \(a\)  | \(b\)  | \(a^2\)   | \(b^2\)   | \(a^2 - b^2\)         |
|------------------------|--------|--------|-----------|-----------|-----------------------|
| \( (2x + 5)(2x - 5) \)  | \(2x\) | \(5\)  | \(4x^2\)  | \(25\)    | \(4x^2 - 25\)         |
| \( (x + 4)(x - 4) \)    | \(x\)  | \(4\)  | \(x^2\)   | \(16\)    | \(x^2 - 16\)          |
| \( (4x + 3)(4x - 3) \)  | \(4x\) | \(3\)  | \(16x^2\) | \(9\)     | \(16x^2 - 9\)         |

---

Would you like more detailed explanations for any of these expansions or formulas? Here are five related questions to deepen understanding:

1. How do the signs in binomial expansions affect the middle term in the result?
2. Can you apply the binomial expansion to any two terms, or do they need to follow specific patterns?
3. What is the geometric interpretation of \( (a + b)(a - b) = a^2 - b^2 \)?
4. How do the coefficients in \( a^2 \), \( 2ab \), and \( b^2 \) change as you increase the values of \( a \) and \( b \)?
5. How would the expansion change if we cube binomials instead of squaring them?

**Tip**: Always remember that the middle term in a binomial expansion reflects the product of both terms, doubled if it is a sum or subtracted twice if it is a difference.

Complete the tables with expansions of binomials (a+b)^2, (a-b)^2, and (a+b)(a-b) provided in the image.

Learn how to expand binomials using formulas for (a+b)^2, (a-b)^2, and (a+b)(a-b). This example includes detailed steps for expressions such as (3x+2)^2 and (x-5)^2, demonstrating key concepts in quadratic equations.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation