| Exam Board | Edexcel |
|---|---|
| Module | C1 (Core Mathematics 1) |
| Marks | 6 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Completing the square and sketching |
| Type | Complete square then solve equation |
| Difficulty | Moderate -0.3 This is a straightforward C1 completing the square question with standard algebraic manipulation. Part (a) requires routine completion of the square with a parameter k, and part (b) is direct substitution. While it involves multiple steps and algebraic fluency, it's a textbook exercise requiring no problem-solving insight, making it slightly 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 |
|---|---|---|
| (a) \((x + k)^2 - k^2 + 4 = 0\) | M1 | |
| \((x + k)^2 = k^2 - 4\) | A1 | |
| \(x + k = \pm\sqrt{k^2 - 4}\) | M1 | |
| \(x = -k \pm \sqrt{k^2 - 4}\) | A1 | |
| (b) \(k = 3 \therefore x = -3 \pm \sqrt{3^2 - 4}\) | M1 | |
| \(= -3 \pm \sqrt{5}\) | A1 | (6) |
**(a)** $(x + k)^2 - k^2 + 4 = 0$ | M1 |
$(x + k)^2 = k^2 - 4$ | A1 |
$x + k = \pm\sqrt{k^2 - 4}$ | M1 |
$x = -k \pm \sqrt{k^2 - 4}$ | A1 |
**(b)** $k = 3 \therefore x = -3 \pm \sqrt{3^2 - 4}$ | M1 |
$= -3 \pm \sqrt{5}$ | A1 | (6)\begin{enumerate}[label=(\alph*)]
\item By completing the square, find in terms of the constant $k$ the roots of the equation
$$x^2 + 2kx + 4 = 0.$$ [4]
\item Hence find the exact roots of the equation
$$x^2 + 6x + 4 = 0.$$ [2]
\end{enumerate}
\hfill \mbox{\textit{Edexcel C1 Q5 [6]}}