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.