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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.

# Summing to One

##### Age 16 to 18Challenge Level

This resource is from Underground Mathematics.

$$\log_3 3=1$$

$$\log_9 3+ \log_9 3 =1$$

$$\log_{27} 3 + \dots = 1$$

How many $\log_{81} 3$ do you need to add together to make one?

Can we choose integers $x$ and $y$ so that:

$$\log_6 x +\log_6 y =1?$$

How many different ways are there to do this?

How about using $\log_{12}$?

How about using $\log_{24}$?

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.