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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

This resource is from Underground Mathematics.

$$ax^2 + bx + c = 0,$$
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.

To show that a statement is ALWAYS true, we need to give a proof.

To show that a statement is NEVER true, we need to give a proof.

To show that a statement is SOMETIMES true, we need to give an example
when it is true and an example when it is false.  If you want a harder
challenge, can you determine exactly when it is and when it is not
true?

You might want to print and cut out the statements (downloadable from
the link here), so that you can sort them into piles.

(1) If $a < 0$, then the equation has no real roots

(2) If $b^2 - 4ac = 0$, then the equation has one repeated real root.

(3) If the equation has no real roots, then the equation $ax^2 + bx - c = 0$ has two distinct real roots.

(4) If $\frac{b^2}{a} < 4c$, then the equation has two distinct real roots.

(5) If $b = 0$, then the equation has one repeated real root.

(6) The equation has three real roots.

(7) If $c = 0$, then the equation has no real roots.

(8) The equation has the same number of real roots as $ax^2 - bx + c = 0$.

(9) If the equation has two distinct real roots, then $ac < \frac{b^2}{4}$.

(10) If $c > 0$, then the equation has two distinct real roots.

(11) The equation has the same number of real roots as the equation $cx^2 + bx + a = 0$.

(12) If the equation has no real roots, then the equation $-ax^2 - bx - c = 0$ has two distinct real roots.

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.