A LevelMaths: PureAQARevision Notes2. Algebra & Functions2.2 Quadratics2.2.5 Further Solving Quadratic Equations (Hidden Quadratics)

Further Solving Quadratic Equations (Hidden Quadratics) (AQA A Level Maths: Pure)

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Paul

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8 April 2023

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Further Solving Quadratic Equations (Hidden Quadratics)

What are hidden quadratic equations?

Hidden quadratic equations are quadratics written in terms of a function $f (x)$
A normal quadratic appears in the form $a x^{2} + b x + c = 0$
Whereas a hidden quadratic appears in the form $a {[f (x)]}^{2} + b [f (x)] + c = 0$
This might look complicated but it simply means $x$ has been replaced by $f (x)$
e.g. $\sin^{2} x + 2 \sin x - 3 = 0$ is just the hidden quadratic of $x^{2} + 2 x - 3 = 0$ where $f (x) = \sin x$

How to solve hidden quadratic equations

First rearrange the function into the form $a {[f (x)]}^{2} + b [f (x)] + c = 0$
Replacing the function and solving the 'normal' quadratic first
Then substitute the function back into the solutions to solve the original quadratic

Worked example

Further Solving of Quadratic Equations Example, A Level & AS Level Pure Maths Revision Notes

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Author: Paul

Expertise: Maths

Paul has taught mathematics for 20 years and has been an examiner for Edexcel for over a decade. GCSE, A level, pure, mechanics, statistics, discrete – if it’s in a Maths exam, Paul will know about it. Paul is a passionate fan of clear and colourful notes with fascinating diagrams – one of the many reasons he is excited to be a member of the SME team.