https://file.aiquickdraw.com/mathbot/file/1729566719976530hrg1i.jpg

Let's solve the equations from the image step by step.

### 1. \( a \)
#### (i) Solve \( x^2 + 5x = 0 \)

This is a quadratic equation. We can factor it as:
\[
x(x + 5) = 0
\]
Set each factor equal to 0:
\[
x = 0 \quad \text{or} \quad x + 5 = 0
\]
\[
x = 0 \quad \text{or} \quad x = -5
\]

So, the solutions are \( x = 0 \) and \( x = -5 \).

#### (ii) Solve \( x^2 - 3x = 0 \)

Factor the equation:
\[
x(x - 3) = 0
\]
Set each factor equal to 0:
\[
x = 0 \quad \text{or} \quad x - 3 = 0
\]
\[
x = 0 \quad \text{or} \quad x = 3
\]

So, the solutions are \( x = 0 \) and \( x = 3 \).

---

### 1. \( b \)
#### (i) Solve \( 4x^2 = 12x \)

First, move all terms to one side:
\[
4x^2 - 12x = 0
\]
Factor out the common factor:
\[
4x(x - 3) = 0
\]
Set each factor equal to 0:
\[
4x = 0 \quad \text{or} \quad x - 3 = 0
\]
\[
x = 0 \quad \text{or} \quad x = 3
\]

So, the solutions are \( x = 0 \) and \( x = 3 \).

#### (ii) Solve \( 5x^2 = 20x \)

First, move all terms to one side:
\[
5x^2 - 20x = 0
\]
Factor out the common factor:
\[
5x(x - 4) = 0
\]
Set each factor equal to 0:
\[
5x = 0 \quad \text{or} \quad x - 4 = 0
\]
\[
x = 0 \quad \text{or} \quad x = 4
\]

So, the solutions are \( x = 0 \) and \( x = 4 \).

---

Let me know if you'd like more details or have any questions. Here are 5 questions that build on this:

1. How do we recognize when an equation can be factored?
2. Why is factoring a useful method for solving quadratic equations?
3. What are other methods for solving quadratic equations apart from factoring?
4. What is the significance of a solution being \( x = 0 \)?
5. Can quadratic equations have complex solutions?

**Tip:** Always check your factored terms by expanding them to ensure you factored the equation correctly.

Solve these quadratic equations: (a) i) x^2 + 5x = 0, ii) x^2 - 3x = 0; (b) i) 4x^2 = 12x, ii) 5x^2 = 20x.

This guide explains how to solve quadratic equations by factoring. Example problems include x^2 + 5x = 0 and 4x^2 = 12x. Key algebraic concepts such as factoring and the Zero Product Property are covered. Suitable for students in Grades 9-11.

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