Solving Hidden Quadratics using Substitutions (Cambridge (CIE) A Level Maths: Pure 1): Revision Note

Solving hidden quadratics using substitutions

What are hidden quadratic equations?

• Hidden quadratic equations are quadratics written in terms of a function $f\left(x\right)$

• A normal quadratic appears in the form $a{x}^2+bx+c=0$

• Whereas a hidden quadratic appears in the form $a{\left[f\left(x\right)\right]}^2+b\left[f\left(x\right)\right]+c=0$

• This might look complicated but it simply means $x$ has been replaced by $f\left(x\right)$

• e.g. ${\sin }^{2}x+2\sin x-3=0$ is just the hidden quadratic of $x^2+2x-3=0$ where $f\left(x\right)=\sin x$

How do I solve hidden quadratic equations?

• First rearrange the function into the form $a{\left[f\left(x\right)\right]}^2+b\left[f\left(x\right)\right]+c=0$

• Replacing the function and solving the 'normal' quadratic first

• Then substitute the function back into the solutions to solve the original quadratic