1. Let's explore a higher-level GCSE topic: quadratic equations. 2. Problem: Solve the quadratic equation $x^2 - 5x + 6 = 0$. 3. Formula and rules: Quadratic equations can be solved by factoring, completing the square, or using the quadratic formula: $$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$$ where $a$, $b$, and $c$ are coefficients from $ax^2 + bx + c = 0$. 4. Step 1: Identify coefficients: $a=1$, $b=-5$, $c=6$. 5. Step 2: Calculate the discriminant: $$\Delta = b^2 - 4ac = (-5)^2 - 4 \times 1 \times 6 = 25 - 24 = 1$$ 6. Step 3: Since $\Delta > 0$, there are two real roots. Use the quadratic formula: $$x = \frac{-(-5) \pm \sqrt{1}}{2 \times 1} = \frac{5 \pm 1}{2}$$ 7. Step 4: Calculate the two solutions: $$x_1 = \frac{5 + 1}{2} = 3$$ $$x_2 = \frac{5 - 1}{2} = 2$$ 8. Explanation: We used the quadratic formula to find the roots of the equation by calculating the discriminant and then substituting values. 9. Final answer: $x = 3$ or $x = 2$. This topic is important for higher GCSE math and helps in understanding polynomial equations.