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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.

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.

# Discriminating

##### Age 16 to 18
Below are several statements about the quadratic equation
$$ax^2 + bx + c = 0,$$
where $a$, $b$ and $c$ are allowed to be any real numbers except that $a$ is not $0$.

For each statement, decide whether it is ALWAYS true, SOMETIMES true, or NEVER true.

Do you have (or can you come up with) any favourite examples of quadratics with different numbers of real roots?  Having some examples to test is a good way to tackle a question like this.

How can we use the coefficients of a quadratic equation to tell us about its number of real roots?  How does the quadratic formula come into this?
This is an Underground Mathematics resource