1. **Problem statement:** We have a trapezium with top base $x+7$, bottom base $2x+1$, height $x$, and area 42 cm². 2. **Formula for area of trapezium:** $$\text{Area} = \frac{(\text{top base} + \text{bottom base}) \times \text{height}}{2}$$ 3. **Substitute given values:** $$42 = \frac{(x+7 + 2x+1) \times x}{2}$$ 4. **Simplify inside the parentheses:** $$42 = \frac{(3x + 8) \times x}{2}$$ 5. **Multiply both sides by 2:** $$84 = (3x + 8) x$$ 6. **Expand the right side:** $$84 = 3x^2 + 8x$$ 7. **Bring all terms to one side:** $$3x^2 + 8x - 84 = 0$$ This proves part (a). --- 8. **Solve the quadratic equation:** $$3x^2 + 8x - 84 = 0$$ 9. **Use the quadratic formula:** $$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$$ where $a=3$, $b=8$, $c=-84$. 10. **Calculate the discriminant:** $$\Delta = 8^2 - 4 \times 3 \times (-84) = 64 + 1008 = 1072$$ 11. **Calculate the square root:** $$\sqrt{1072} \approx 32.75$$ 12. **Find the two roots:** $$x = \frac{-8 \pm 32.75}{6}$$ 13. **Calculate each root:** - Positive root: $$x = \frac{-8 + 32.75}{6} = \frac{24.75}{6} = 4.13$$ - Negative root: $$x = \frac{-8 - 32.75}{6} = \frac{-40.75}{6} = -6.79$$ 14. **Interpretation:** Height cannot be negative, so the height is approximately $4.13$ cm. **Final answer:** The height of the trapezium is $4.13$ cm (to 2 decimal places).