| Exam Board | Edexcel |
|---|---|
| Module | C1 (Core Mathematics 1) |
| Marks | 6 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Completing the square and sketching |
| Type | Discriminant for real roots condition |
| Difficulty | Moderate -0.8 Part (a) requires routine discriminant calculation or recognizing a perfect square (b²-4ac=0), which is straightforward recall. Part (b) involves completing the square and solving a simple quadratic, but the surd form requirement adds minimal complexity. This is a standard C1 exercise with clear methods and no problem-solving insight needed, making it easier than average. |
| Spec | 1.02d Quadratic functions: graphs and discriminant conditions1.02f Solve quadratic equations: including in a function of unknown |
| Answer | Marks | Guidance |
|---|---|---|
| (a) \(b^2 - 4ac = 12^2 - (4 \times 4 \times 9) = 0\) | M1 | |
| \(\therefore\) 1 real root | A1 | |
| (b) \(4x^2 + 12x + 9 = 8\) | ||
| \(4x^2 + 12x + 1 = 0\) | ||
| \(x = \frac{-12 \pm \sqrt{144-16}}{8}\) | M1 | |
| \(= \frac{-12 \pm 8\sqrt{2}}{8}\) | M1 | |
| \(= -\frac{3}{2} \pm \sqrt{2}\) | A2 | (6) |
**(a)** $b^2 - 4ac = 12^2 - (4 \times 4 \times 9) = 0$ | M1 |
$\therefore$ 1 real root | A1 |
**(b)** $4x^2 + 12x + 9 = 8$ | |
$4x^2 + 12x + 1 = 0$ | |
$x = \frac{-12 \pm \sqrt{144-16}}{8}$ | M1 |
$= \frac{-12 \pm 8\sqrt{2}}{8}$ | M1 |
$= -\frac{3}{2} \pm \sqrt{2}$ | A2 | (6)$\text{f}(x) = 4x^2 + 12x + 9$.
\begin{enumerate}[label=(\alph*)]
\item Determine the number of real roots that exist for the equation $\text{f}(x) = 0$. [2]
\item Solve the equation $\text{f}(x) = 8$, giving your answers in the form $a + b\sqrt{2}$ where $a$ and $b$ are rational. [4]
\end{enumerate}
\hfill \mbox{\textit{Edexcel C1 Q3 [6]}}