June 2017 Paper 2 Q24
24 Write \(\quad 12x^2 - 60x + 5 \quad\) in the form \(\quad a(bx + c)^2 + d \quad\) where \(a\), \(b\), \(c\) and \(d\) are integers. [5 marks]
| Answer | Mark | Comments |
|---|---|---|
| Alternative method 1 | ||
| \(12(x^2 - 5x)\ \ldots\) or \(12(x - 2.5)^2\ \ldots\) | M1 | oe eg \(12\{(x^2 - 5x)\ \ldots\}\) or \(12(x^2 - 5x\ \ldots)\) |
| \(12\{(x - 2.5)^2 - 2.5^2\}\ldots\) or \(12(x - 2.5)^2 - 75\ \ldots\) | M1dep | oe eg \(12\{(x - 2.5)^2 - 2.5^2\ldots\}\) |
| \(12(x - 2.5)^2 - 12 \times 2.5^2 + 5\) or \(12(x - 2.5)^2 - 70\) | M1dep | oe eg \(12(x - 2.5)^2 - 12 \times 2.5^2 + 12 \times \dfrac{5}{12}\) |
| \(12\left(\dfrac{2x - 5}{2}\right)^2 - 12 \times 2.5^2 + 5\) | M1dep | oe eg \(12\left(\dfrac{2x - 5}{2}\right)^2 - 12 \times 2.5^2 + 12 \times \dfrac{5}{12}\) |
| \(3(2x - 5)^2 - 70\) or \(a = 3 \quad b = 2 \quad c = -5 \quad d = -70\) or \(3(5 - 2x)^2 - 70\) or \(a = 3 \quad b = -2 \quad c = 5 \quad d = -70\) | A1 | oe |
| Alternative method 2 | ||
| \(3(4x^2 - 20x)\ \ldots\) or \(3(2x - 5)^2\ \ldots\) | M1 | oe eg \(3\{(4x^2 - 20x)\ \ldots\}\) or \(3(4x^2 - 20x\ \ldots)\) |
| \(3\{(2x - 5)^2 - 5^2\}\ \ldots\) or \(3(2x - 5)^2 - 75\ \ldots\) | M1dep | oe eg \(3\{(2x - 5)^2 - 5^2\ldots\}\) |
| \(3\{(2x - 5)^2 - 5^2\} + 5\) | M1dep | oe eg \(3\left\{(2x - 5)^2 - 5^2 + \dfrac{5}{3}\right\}\) |
| \(3(2x - 5)^2 - 3 \times 5^2 + 5\) | M1dep | oe eg \(3(2x - 5)^2 - 3 \times 5^2 + 3 \times \dfrac{5}{3}\) |
| \(3(2x - 5)^2 - 70\) or \(a = 3 \quad b = 2 \quad c = -5 \quad d = -70\) or \(3(5 - 2x)^2 - 70\) or \(a = 3 \quad b = -2 \quad c = 5 \quad d = -70\) | A1 | oe |
Additional guidance
| For M marks 2.5 may be seen as \(\dfrac{5}{2}\) | |
| For M marks \((x - 2.5)^2\) may be replaced by \((2.5 - x)^2\) etc | |
| Expansion of given form followed by trial and improvement eg1 \(3(2x - 5)^2 - 70\) (or \(a = 3 \quad b = 2 \quad c = -5 \quad d = -70\)) eg2 Not fully correct | 5 marks Zero |