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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.

# A Tangent Is ...

##### Age 16 to 18Challenge Level

This resource is from Underground Mathematics.

Here are some suggestions of how we could define a tangent to a curve
at a point.

1.  "A tangent is a straight line which only meets the curve at that
one point."

2.  "A tangent is a straight line which touches the curve at that
point only."

3.  "A tangent is a straight line which meets the curve at that point,
but the curve is all on one side of the line."

4.  "A tangent is a straight line which meets the curve at that point,
but near that point, the curve is all on one side of the line."

For each one, can you find or sketch an example to show that the
proposed definition does not always work?

Can you come up with a better definition?

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.