Question: Solve $x^2 - 5x + 6 = 0$ using factorisation. Create a video showing every step on a virtual whiteboard, with English voice narration. Explain why each step is necessary, and verify the answers at the end.
1. **State the problem:** We need to solve the quadratic equation $$x^2 - 5x + 6 = 0$$ using factorisation.
2. **Recall the factorisation method:** For a quadratic equation in the form $$ax^2 + bx + c = 0$$, we look for two numbers that multiply to $$ac$$ and add to $$b$$.
3. **Identify coefficients:** Here, $$a = 1$$, $$b = -5$$, and $$c = 6$$.
4. **Find two numbers:** We want two numbers that multiply to $$1 \times 6 = 6$$ and add to $$-5$$. These numbers are $$-2$$ and $$-3$$ because $$-2 \times -3 = 6$$ and $$-2 + -3 = -5$$.
5. **Rewrite the middle term:** Express $$-5x$$ as $$-2x - 3x$$:
$$x^2 - 2x - 3x + 6 = 0$$
6. **Group terms:** Group the terms to factor by grouping:
$$(x^2 - 2x) + (-3x + 6) = 0$$
7. **Factor each group:**
$$x(x - 2) - 3(x - 2) = 0$$
8. **Factor out the common binomial:**
$$(x - 3)(x - 2) = 0$$
9. **Apply zero product property:** Set each factor equal to zero:
$$x - 3 = 0 \implies x = 3$$
$$x - 2 = 0 \implies x = 2$$
10. **Verify solutions:** Substitute back into the original equation:
- For $$x=3$$: $$3^2 - 5(3) + 6 = 9 - 15 + 6 = 0$$
- For $$x=2$$: $$2^2 - 5(2) + 6 = 4 - 10 + 6 = 0$$
Both satisfy the equation, so the solutions are correct.
**Final answer:** $$x = 2$$ or $$x = 3$$.