Higher June 2025 Paper 4 Q23
23
(a) Write this expression in its simplest factorised form.
\((x + 4)(x - 4)^2 - x^2(x - 5)\)
You must show your working. [6]
(b) Use your answer to part (a) to write down the value of \(x\) which gives the smallest value of this expression. [1]
| Answer | Marks | Part marks and guidance | |
|---|---|---|---|
| \((x - 8)^2\) or \((x - 8)(x - 8)\) final answer with correct working | 6 | B5 for answer \(x^2 - 16x + 64\) with correct working | correct working requires at least M3 or M1 M1 M1 or M2 M1 If you award B marks do not award any extra M marks |
| OR B4 for answer \(-9x^2 - 16x + 64\) with correct working | |||
| OR M3 for correctly expanding the triple bracket e.g. \(x^3 - 4x^2 - 16x + 64\) oe | e.g. \(x^3 - 8x^2 + 16x + 4x^2 - 32x + 64\) oe | ||
| or M2 for correctly expanding any two brackets from the triple bracket e.g. \(x^2\ [-4x + 4x] - 16\) or \(x^2 - 4x - 4x + 16\) oe or \((x^2 - 8x + 16)(x + 4)\) | e.g. \(x^2 - 16\) or \(x^2 - 8x + 16\) condone one error in the previous step e.g \((x^2 - 8x + 8)(x + 4) = x^3 + 4x^2 - 8x^2 - 32x + 8x + 32\) or better scores M2 | ||
| or M1 for attempting to expand any two brackets e.g. \(x^2 + 4x - 4x - 16\) oe with at most one error | e.g. \(x^2 - 8\) | ||
| AND M1 for correctly expanding final bracket e.g. \(-x^3 + 5x^2\) oe | Note : brackets must be removed and \((-x^3 + 5x^2)\) must not be used for multiplication | ||
| AND M1 for correctly simplifying their final expression by collecting like terms e.g. \(x^2 - 16x + 64\) | |||
| If 0,1 or 2 scored instead SC3 for answer \((x - 8)^2\) or \((x - 8)(x - 8)\) | Note : once mark(s) are awarded they are not lost for subsequent incorrect working | ||
| Answer | Marks | Part marks and guidance | |
|---|---|---|---|
| 8 | 1FT | Note : correct answer 8 scores 1 FT their \((ax^2 + bx + c)\) where \(a \gt 0\) and their answer should be \(-\frac{b}{2a}\). | If the quadratic expression is factorised, expand the brackets first. |