| Exam Board | Edexcel |
|---|---|
| Module | C1 (Core Mathematics 1) |
| Year | 2005 |
| Session | June |
| Marks | 6 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Completing the square and sketching |
| Type | Complete square then solve equation |
| Difficulty | Moderate -0.8 This is a straightforward completing the square exercise with routine algebraic manipulation. Part (a) requires standard completion of the square (a=-4, b=-45), and part (b) is a direct application to find roots in surd form. The question guides students through each step and requires only basic algebraic techniques with no problem-solving insight, making it easier than average. |
| Spec | 1.02e Complete the square: quadratic polynomials and turning points1.02f Solve quadratic equations: including in a function of unknown |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| \(x^2 - 8x - 29 \equiv (x-4)^2 - 45\) | M1 | M1 for \((x \pm 4)^2\) or equation for \(a\) |
| \((x-4)^2 - 16 + (-29)\) | A1 | |
| \((x \pm 4)^2 - 45\) | A1 (3) | |
| ALT: Compare coefficients: \(-8 = 2a\), equation for \(a\); \(a = -4\) AND \(a^2 + b = -29\); \(b = -45\) | M1 A1 A1 (3) |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| \((x-4)^2 = 45\) (follow through their \(a\) and \(b\)) | M1 | M1 for full method leading to \(x - 4 = \ldots\) or \(x = \ldots\) |
| \(\Rightarrow x - 4 = \pm\sqrt{45}\), so \(c = 4\) | A1 | A1 for \(c\) |
| \(x = 4 \pm 3\sqrt{5}\), so \(d = 3\) | A1 (3) | A1 for \(d\); use of formula ending with \(\frac{8 \pm 6\sqrt{5}}{2}\) scores M1 A1 A0 |
## Question 3:
### Part (a)
| Answer | Mark | Guidance |
|--------|------|----------|
| $x^2 - 8x - 29 \equiv (x-4)^2 - 45$ | M1 | M1 for $(x \pm 4)^2$ or equation for $a$ |
| $(x-4)^2 - 16 + (-29)$ | A1 | |
| $(x \pm 4)^2 - 45$ | A1 (3) | |
**ALT:** Compare coefficients: $-8 = 2a$, equation for $a$; $a = -4$ AND $a^2 + b = -29$; $b = -45$ | M1 A1 A1 (3)
### Part (b)
| Answer | Mark | Guidance |
|--------|------|----------|
| $(x-4)^2 = 45$ (follow through their $a$ and $b$) | M1 | M1 for full method leading to $x - 4 = \ldots$ or $x = \ldots$ |
| $\Rightarrow x - 4 = \pm\sqrt{45}$, so $c = 4$ | A1 | A1 for $c$ |
| $x = 4 \pm 3\sqrt{5}$, so $d = 3$ | A1 (3) | A1 for $d$; use of formula ending with $\frac{8 \pm 6\sqrt{5}}{2}$ scores M1 A1 A0 |
---$$x ^ { 2 } - 8 x - 29 \equiv ( x + a ) ^ { 2 } + b ,$$
where $a$ and $b$ are constants.
\begin{enumerate}[label=(\alph*)]
\item Find the value of $a$ and the value of $b$.
\item Hence, or otherwise, show that the roots of
$$x ^ { 2 } - 8 x - 29 = 0$$
are $c \pm d \sqrt { } 5$, where $c$ and $d$ are integers to be found.
\end{enumerate}
\hfill \mbox{\textit{Edexcel C1 2005 Q3 [6]}}