Quadratic in higher integer powers

A question is this type if and only if it requires solving an equation of the form ax^(2n) + bx^n + c = 0 where n >= 2 is a positive integer (e.g. x^4, x^6), treated as a quadratic in x^n.

10 questions · Moderate -0.3

1.02f Solve quadratic equations: including in a function of unknown
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OCR C1 2008 January Q10
10 marks Standard +0.3
10 Given that \(\mathrm { f } ( x ) = 8 x ^ { 3 } + \frac { 1 } { x ^ { 3 } }\),
  1. find \(\mathrm { f } ^ { \prime \prime } ( x )\),
  2. solve the equation \(\mathrm { f } ( x ) = - 9\).
OCR C1 2005 June Q4
5 marks Standard +0.3
4 Solve the equation \(x ^ { 6 } + 26 x ^ { 3 } - 27 = 0\).
OCR MEI C1 2008 June Q9
4 marks Moderate -0.8
9 Solve the equation \(y ^ { 2 } - 7 y + 12 = 0\).
Hence solve the equation \(x ^ { 4 } - 7 x ^ { 2 } + 12 = 0\). Section B (36 marks)
OCR C1 Q2
4 marks Moderate -0.3
2. Find in exact form the real solutions of the equation $$x ^ { 4 } = 5 x ^ { 2 } + 14 .$$
OCR C1 2010 June Q5
5 marks Moderate -0.8
5 Find the real roots of the equation \(4 x ^ { 4 } + 3 x ^ { 2 } - 1 = 0\).
OCR H240/01 2018 June Q3
4 marks Moderate -0.8
3 In this question you must show detailed reasoning.
Find the two real roots of the equation \(x ^ { 4 } - 5 = 4 x ^ { 2 }\). Give the roots in an exact form.
CAIE P1 2023 June Q4
3 marks Standard +0.3
Solve the equation \(8x^6 + 215x^3 - 27 = 0\). [3]
OCR C1 2013 June Q2
5 marks Standard +0.3
Solve the equation \(8x^6 + 7x^3 - 1 = 0\). [5]
OCR MEI C1 Q3
4 marks Moderate -0.8
Solve the equation \(y^2 - 7y + 12 = 0\). Hence solve the equation \(x^4 - 7x^2 + 12 = 0\). [4]
OCR PURE Q6
11 marks Moderate -0.3
  1. Determine the two real roots of the equation \(8x^6 + 7x^3 - 1 = 0\). [3]
  2. Determine the coordinates of the stationary points on the curve \(y = 8x^7 + \frac{49}{4}x^4 - 7x\). [4]
  3. For each of the stationary points, use the value of \(\frac{d^2y}{dx^2}\) to determine whether it is a maximum or a minimum. [4]