June 2025 Paper 1 Q11
11 Solve \(\quad x^3 + 10x^2 + 8x = 0\)
Where appropriate give your answers in surd form.
Simplify your answers where possible. [4 marks]
| Answer | Mark | Comments |
|---|---|---|
| Alternative method 1: quadratic formula | ||
| \(x(x^2 + 10x + 8)\ \ (= 0)\) or \(x = 0\) | M1 | |
| \((x =)\ \dfrac{-10 \pm \sqrt{\left(10^2 - 4(\times 1) \times 8\right)}}{2(\times 1)}\) | M1 | oe correct substitution into formula from correct quadratic |
| \((x =)\ \dfrac{-10 \pm \sqrt{100 - 32}}{2}\) or \(\dfrac{-10 \pm \sqrt{68}}{2}\) | A1 | oe simplification eg \(-5 \pm \dfrac{\sqrt{68}}{2}\) dep on second M mark only implied by final A1 |
| \((x =)\ 0,\ -5 \pm \sqrt{17}\) | A1 | could be written as two separate surds. |
| Alternative method 2: completes the square | ||
| \(x(x^2 + 10x + 8)\ \ (= 0)\) or \(x = 0\) | M1 | |
| \(x\left[(x + 5)^2 \ldots\ldots\right]\) | M1 | oe eg may not include the common factor of \(x\) at this stage must be from correct quadratic |
| \(x\left[(x + 5)^2 - 17\right]\) | A1 | oe eg \((x + 5)^2 = 17\) dep on second M mark only condone common factor of \(x\) not being there at this stage |
| \((x =)\ 0,\ -5 \pm \sqrt{17}\) | A1 | could be written as two separate surds. |
Additional guidance
| If \(x = 0\) stated as final answer then it doesn’t need to appear in the working | |
| If \(x = 0\) not stated on answer line then second A mark cannot be awarded: eg \(x = 0\) is seen early in the working with everything else correct \(x = 0\) or \(x(x^2 + 10x + 8)\) not seen in the working but with everything else correct | M2A1A0 M0M1A1A0 |
| Factor theorem used correctly to get complete answer | M2A2 |
| Factor theorem used without complete answer | M0A0 |