November 2021 Paper 2 Q21
21 \(3x^2 + 2bx + 8a \quad\) can be written in the form \(\quad 3(x + a)^2 + b + 2\)
Work out the two possible pairs of values of \(a\) and \(b\). [6 marks]
| Answer | Mark | Comments |
|---|---|---|
| Alternative method 1 | ||
| \(3(x^2 + ax + ax + a^2) \ldots\) or \(3(x^2 + 2ax + a^2) \ldots\) or \(3\left(x + \dfrac{b}{3}\right)^2 \ldots\) or \(2b = 6a\) or \(8a = 3a^2 + b + 2\) | M1 | oe eg \(3x^2 + 6ax + 3a^2 \ldots\) or \(\dfrac{b}{3} = a\) or \(b + 2 = -3\left(\dfrac{b}{3}\right)^2 + 8a\) |
| \(2b = 6a\) and \(8a = 3a^2 + b + 2\) | M1dep | oe equations eg \(\dfrac{b}{3} = a\) and \(b + 2 = -3\left(\dfrac{b}{3}\right)^2 + 8a\) |
| \(3a^2 + 3a - 8a + 2\ (= 0)\) or \(3a^2 - 5a + 2\ (= 0)\) | M1dep | oe quadratic equation in \(a\) |
| \((3a - 2)(a - 1)\) or \(\dfrac{--5 \pm \sqrt{(-5)^2 - 4 \times 3 \times 2}}{2 \times 3}\) | M1 | oe eg \(\dfrac{5}{6} \pm \sqrt{\dfrac{25}{36} - \dfrac{2}{3}}\) ft their 3-term quadratic |
| \(a = \dfrac{2}{3}\) and \(a = 1\) or \(a = \dfrac{2}{3}\) and \(b = 2\) or \(a = 1\) and \(b = 3\) | A1 | |
| \(a = \dfrac{2}{3}\) and \(b = 2\) and \(a = 1\) and \(b = 3\) | A1 | |
| Alternative method 2 | ||
| \(3(x^2 + ax + ax + a^2) \ldots\) or \(3(x^2 + 2ax + a^2) \ldots\) or \(3\left(x + \dfrac{b}{3}\right)^2 \ldots\) or \(2b = 6a\) or \(8a = 3a^2 + b + 2\) | M1 | oe eg \(3x^2 + 6ax + 3a^2 \ldots\) or \(\dfrac{b}{3} = a\) or \(b + 2 = -3\left(\dfrac{b}{3}\right)^2 + 8a\) |
| \(2b = 6a\) and \(8a = 3a^2 + b + 2\) | M1dep | oe equations eg \(\dfrac{b}{3} = a\) and \(b + 2 = -3\left(\dfrac{b}{3}\right)^2 + 8a\) |
| \(\dfrac{8b}{3} = 3\left(\dfrac{b}{3}\right)^2 + b + 2\) or \(b^2 - 5b + 6\ (= 0)\) | M1dep | oe quadratic equation in \(b\) |
| \((b - 2)(b - 3)\) or \(\dfrac{--5 \pm \sqrt{(-5)^2 - 4 \times 1 \times 6}}{2 \times 1}\) | M1 | oe eg \(\dfrac{5}{2} \pm \sqrt{\dfrac{25}{4} - 6}\) ft their 3-term quadratic |
| \(b = 2\) and \(b = 3\) or \(a = \dfrac{2}{3}\) and \(b = 2\) or \(a = 1\) and \(b = 3\) | A1 | |
| \(a = \dfrac{2}{3}\) and \(b = 2\) and \(a = 1\) and \(b = 3\) | A1 | |
Additional guidance
Allow \(0.\dot{6}\) for \(\dfrac{2}{3}\)
Allow 0.67 for \(\dfrac{2}{3}\) for first A1
In quadratic formula allow \(5^2\) for \((-5)^2\) but use of \(-5^2\) must be recovered