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This comes in two parts, with the first being less fiendish than the second. Itâ€™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.

### Discriminating

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.

# Integral Chasing

##### Age 16 to 18Challenge Level

This resource is from Underground Mathematics.

Below is a selection of integrals, some of which you can't do directly and some that require the use of other mathematical skills.

Can you find the missing positive numbers $a$ to $d$?

$$\text{(1)} \ \int_a^5 10x+3 \ \text{d}x = 114$$

$$\text{(2)} \ \int_{2a}^9 b\sqrt{x}+\dfrac{a}{\sqrt{x}}\ \text{d}x=42$$

$$\text{(3)} \ \int_{\frac{1}{2}}^1 \dfrac{1}{x^5}-\dfrac{1}{x^2} \ \text{d}x=\dfrac{c+1}{4}$$

$$\text{(4)} \ \int^{c+2}_6 x^{\frac{b}{a}}\left(\sqrt{x}-\dfrac{1}{\sqrt{x}}\right) \ \text{d}x=ab^ad^a$$

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.