A-LevelMathematicsQuadratics9709/12 · Oct/Nov 2025 · Q15 Marks
Express 9x²-36x+8 in the form p(x+q)²+r, where p, q and r are constants.
- (a)Express 9x²-36x+8 in the form p(x+q)²+r, where p, q and r are constants.[2]
- (b)Hence find the set of values of the constant k for which the equation 9x²-36x+8 = k has no real roots.[1]
- (c)Find the exact roots of the equation 9x² - 36x+8=-15.[2]
Mark scheme
Marking points from the official mark scheme (Oct/Nov 2025). The mark or mark type is shown in brackets.
- [B2] p=9, q=-2, r=-28 (B2 for all 3 correct; B1 for any two correct. Allow B1 for 9{(x – 2)² + k}.)
- [B1] k < their r
- [M1] Solving 9(x-2)² - 28 = -15 for x, to the stage (x-2)² = 13/9 or equivalent quadratic formula application.
- [A1] x = 2 ± √13/3 or equivalent exact form (e.g. (6 ± √13)/3, (36 ± √468)/18).
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