https://file.aiquickdraw.com/mathbot/file/1725796454323b479eygq.jpg

The image asks to solve three quadratic equations by factoring. Let me walk you through solving each equation step-by-step.

### 1. \( x^2 + 10x + 25 = 0 \)

Step-by-step factoring:
1. This is a quadratic of the form \( ax^2 + bx + c = 0 \).
2. Notice that \( 25 \) is a perfect square, and \( 10x \) is twice the square root of 25. So this can be factored as:
   \[
   (x + 5)(x + 5) = 0 \quad \text{or} \quad (x + 5)^2 = 0
   \]
3. Solving \( (x + 5)^2 = 0 \), we get:
   \[
   x + 5 = 0 \quad \Rightarrow \quad x = -5
   \]

### 2. \( x^2 + 21 = -10x \)

Step-by-step factoring:
1. First, rearrange the equation to standard form \( ax^2 + bx + c = 0 \):
   \[
   x^2 + 10x + 21 = 0
   \]
2. Now, find two numbers that multiply to give 21 and add to give 10. These numbers are 7 and 3.
3. Factor the quadratic:
   \[
   (x + 7)(x + 3) = 0
   \]
4. Solve for \( x \):
   \[
   x + 7 = 0 \quad \Rightarrow \quad x = -7
   \]
   \[
   x + 3 = 0 \quad \Rightarrow \quad x = -3
   \]

### 3. \( b^2 - 2b = 8 \)

Step-by-step factoring:
1. Rearrange the equation to standard form:
   \[
   b^2 - 2b - 8 = 0
   \]
2. Now, find two numbers that multiply to give -8 and add to give -2. These numbers are -4 and 2.
3. Factor the quadratic:
   \[
   (b - 4)(b + 2) = 0
   \]
4. Solve for \( b \):
   \[
   b - 4 = 0 \quad \Rightarrow \quad b = 4
   \]
   \[
   b + 2 = 0 \quad \Rightarrow \quad b = -2
   \]

---

### Summary of solutions:
1. \( x = -5 \)
2. \( x = -7 \) or \( x = -3 \)
3. \( b = 4 \) or \( b = -2 \)

Would you like any additional details or explanations on these steps?

Here are 5 related questions you could explore:
1. How do you know when a quadratic equation can be factored?
2. Can all quadratic equations be factored, or are there other methods to solve them?
3. What do you do if you cannot find factors easily?
4. How can you verify the solutions to these quadratic equations?
5. What role does the discriminant play in factoring quadratics?

**Tip:** When factoring quadratics, always start by rearranging the equation into the form \( ax^2 + bx + c = 0 \) and then look for pairs of numbers that multiply to \( ac \) and add to \( b \).

Solve the following quadratic equations by factoring. Answer the questions that follow: 1) x^2 + 10x + 25 = 0, 2) x^2 + 21 = -10x, 3) b^2 - 2b = 8

Learn how to solve quadratic equations like x^2 + 10x + 25 = 0, x^2 + 21 = -10x, and b^2 - 2b = 8 by factoring. This guide provides step-by-step solutions, covering key algebraic concepts such as factoring and the zero-product property. Suitable for students in Grades 8-10.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation