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Powerful Quadratics

This comes in two parts, with the first being less fiendish than the second. It’s great for practising both quadratics and laws of indices, and you can get a lot from making sure that you find all the solutions. For a real challenge (requiring a bit more knowledge), you could consider finding the complex solutions.


You're invited to decide whether statements about the number of solutions of a quadratic equation are always, sometimes or never true.

Factorisable Quadratics

This will encourage you to think about whether all quadratics can be factorised and to develop a better understanding of the effect that changing the coefficients has on the factorised form.

Nested Surds

Age 16 to 18
Challenge Level

This resource is from Underground Mathematics.

For each of the statements provided below, determine which non-negative values of $a$, $b$, $c$, and $d$, if any, make the equation true.

These can be attempted in any order but you might find that some statements can help inform your decisions about others.



You can download these statements as a set of cards that can be cut out and considered in any order. 
a)  $\sqrt{a} \times \sqrt{b} = \sqrt{ab}$
b)  $\dfrac{\sqrt{a}}{\sqrt{b}} = \sqrt{\dfrac{a}{b}}$
c)  $\sqrt{23-6\sqrt{6-4\sqrt{2}}}=\sqrt{a}+\sqrt{b}$

d)  $a\sqrt{b}=\sqrt{ab}$
e)  $\dfrac{\sqrt{ab}}{\sqrt{a}+\sqrt{b}}=1$
f)  $\sqrt{a} - \sqrt{b}=\sqrt{a - b}$
g)  $\sqrt{a}+\sqrt{b}=\sqrt{a+b+\sqrt{4ab}}$

h)  $\dfrac{\sqrt{a}+b}{\sqrt{c}+d}=(\sqrt{a}+b)(\sqrt{c}-d)$

i)  $\sqrt{5+2\sqrt{6}}=\sqrt{a}+\sqrt{b}$




This is an Underground Mathematics resource.

Underground Mathematics is hosted by Cambridge Mathematics. The project was originally funded by a grant from the UK Department for Education to provide free web-based resources that support the teaching and learning of post-16 mathematics.

Visit the site at to find more resources, which also offer suggestions, solutions and teacher notes to help with their use in the classroom.