Solve quadratic by substitution

A question is this type if and only if it requires solving an equation by substituting y = x^(1/2), y = x², y = tan(x), or similar, reducing it to a quadratic in y.

3 questions · Moderate -0.4

1.02f Solve quadratic equations: including in a function of unknown
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Edexcel C1 2018 June Q3
6 marks Moderate -0.8
3. $$f ( x ) = x ^ { 2 } - 10 x + 23$$
  1. Express \(\mathrm { f } ( x )\) in the form \(( x + a ) ^ { 2 } + b\), where \(a\) and \(b\) are constants to be found.
  2. Hence, or otherwise, find the exact solutions to the equation $$x ^ { 2 } - 10 x + 23 = 0$$
  3. Use your answer to part (b) to find the larger solution to the equation $$y - 10 y ^ { 0.5 } + 23 = 0$$ Write your solution in the form \(p + q \sqrt { r }\), where \(p , q\) and \(r\) are integers.
OCR C1 2006 January Q7
11 marks Standard +0.3
7
  1. Solve the equation \(x ^ { 2 } - 8 x + 11 = 0\), giving your answers in simplified surd form.
  2. Hence sketch the curve \(y = x ^ { 2 } - 8 x + 11\), labelling the points where the curve crosses the axes.
  3. Solve the equation \(y - 8 y ^ { \frac { 1 } { 2 } } + 11 = 0\), giving your answers in the form \(p \pm q \sqrt { 5 }\).
CAIE P1 2024 June Q1
5 marks Moderate -0.8
  1. Express \(3y^2 - 12y - 15\) in the form \(3(y + a)^2 + b\), where \(a\) and \(b\) are constants. [2]
  2. Hence find the exact solutions of the equation \(3x^4 - 12x^2 - 15 = 0\). [3]