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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.

Irrational Roots

Age 16 to 18
Challenge Level

This resource is from Underground Mathematics.

The roots of quadratic polynomials can be nice, integer values. For example $x^2+4x+3$ has $x=-3$ as a root. However, this is not always the case. You will have encountered many quadratic polynomials with roots that are fractions or even irrational numbers.

Convince yourself that $x=\sqrt{2}$ is a root of the quadratic equation $x^2-2=0$ and that $x=\sqrt{3}$ is a root of the quadratic equation $x^2-3=0$.






  • Can you find a quadratic polynomial with integer coefficients which has $x=1+\sqrt{2}$ as a root?

    What is the other root of this polynomial?

  • What if instead $x=1+\sqrt{3}$ is a root? What would the quadratic polynomial be now?

    What would the other root be this time?

  • Can you generalise your answers to the case where $1+\sqrt{n}$ is a root?

    What about the case where $m+\sqrt{n}$ is a root?


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.