Solving Hidden Quadratics using Substitutions (Edexcel IAL Maths)

Further Solving Quadratic Equations (Hidden Quadratics)

Spec point — 1.5

Solving hidden quadratics using substitutions

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 $ax^{2}+bx+c=0$
• 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• This might look complicated but it simply means $x$ has been replaced by $f(x)$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. $\mathrm{sin}^{2}x+2\mathrm{sin}x-3=0$ is just the hidden quadratic of $x^{2}+2x-3=0$ where $f(x)=\mathrm{sin}x$

How do I 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