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### Number and algebra

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# Powerful Quadratics

**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 undergroundmathematics.org to find more resources, which also offer suggestions, solutions and teacher notes to help with their use in the classroom.*
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### Discriminating

### Factorisable Quadratics

### Which Quadratic?

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Age 16 to 18

(i) Find all real solutions of the equation

$$(x^2âˆ’7x+11)^{(x^2âˆ’11x+30)}=1.$$

(ii) Find all real solutions of the equation

$$(2âˆ’x^2)^{(x^2âˆ’3\sqrt{2}x+4)}=1.$$

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

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.

In this activity you will need to work in a group to connect different representations of quadratics.