Quadratic with surd roots, exact form

A question is this type if and only if it requires solving a quadratic equation and expressing the roots exactly in surd form (e.g. p ± q√r), typically using the quadratic formula.

6 questions · Moderate -0.3

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OCR MEI C1 2015 June Q8
5 marks Moderate -0.8
8 Fig. 8 shows a right-angled triangle with base \(2 x + 1\), height \(h\) and hypotenuse \(3 x\). \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{c55e1f96-670a-4bc3-9e77-92d28769b7f5-2_317_593_1653_543} \captionsetup{labelformat=empty} \caption{Not to scale}
\end{figure} Fig. 8
  1. Show that \(h ^ { 2 } = 5 x ^ { 2 } - 4 x - 1\).
  2. Given that \(h = \sqrt { 7 }\), find the value of \(x\), giving your answer in surd form.
OCR MEI C1 Q11
12 marks Moderate -0.8
11
  1. Multiply out \(( x - p ) ( x - q )\).
  2. You are given that \(p = 2 + \sqrt { 3 }\) and \(q = 2 - \sqrt { 3 }\) are the roots of a quadratic equation. Find \(p + q\) and \(p q\) and hence find the quadratic equation with roots \(x = p\) and \(x = q\).
  3. Solve the quadratic equation \(x ^ { 2 } + 5 x - 7 = 0\) giving the roots exactly.
  4. Show that \(x = 1\) is the only root of the equation \(x ^ { 3 } + 2 x - 3 = 0\).
  5. A quadratic equation \(x ^ { 2 } + r x + s = 0\), where \(r\) and \(s\) are integers, has two roots. One root is \(x = 3 + \sqrt { 5 }\). Without finding \(r\) or \(s\), write down the other root.
OCR MEI C1 2007 January Q7
4 marks Standard +0.8
7 You are given that \(a = \frac { 3 } { 2 } , b = \frac { 9 - \sqrt { 17 } } { 4 }\) and \(c = \frac { 9 + \sqrt { 17 } } { 4 }\). Show that \(a + b + c = a b c\).
OCR MEI C4 Q5
Standard +0.8
5 Justify the statement in line 87 that $$\frac { 1 } { \phi } = \frac { \sqrt { 5 } - 1 } { 2 }$$
Pre-U Pre-U 9794/1 2013 November Q4
Moderate -0.8
4 Solve the equation \(x ^ { 2 } + ( \sqrt { 3 } ) x - 18 = 0\), giving each root in the form \(p \sqrt { q }\) where \(p\) and \(q\) are integers.
OCR C1 2013 January Q1
5 marks Moderate -0.8
  1. Solve the equation \(x^2 - 6x - 2 = 0\), giving your answers in simplified surd form. [3]
  2. Find the gradient of the curve \(y = x^2 - 6x - 2\) at the point where \(x = -5\). [2]