### Golden Thoughts

Rectangle PQRS has X and Y on the edges. Triangles PQY, YRX and XSP have equal areas. Prove X and Y divide the sides of PQRS in the golden ratio.

### Roots and Coefficients

If xyz = 1 and x+y+z =1/x + 1/y + 1/z show that at least one of these numbers must be 1. Now for the complexity! When are the other numbers real and when are they complex?

### Two Cubes

Two cubes, each with integral side lengths, have a combined volume equal to the total of the lengths of their edges. How big are the cubes? [If you find a result by 'trial and error' you'll need to prove you have found all possible solutions.]

# Proof Sorter - Quadratic Equation

##### Stage: 4 and 5 Challenge Level:

Why do this problem?

This is one of a series of jumbled proofs. We think that sorting the steps of the proof and the reasons, into the right order will help people to understand and remember the derivation of the formula for the roots of a quadratic equation.

Possible approach

We hope this activity will be particularly useful in pair or group work, where interpretation of the text can be shared and the justification of choices can be tested on fellow students.

Key questions

Which blue statement looks like a starter?

Which line of algebra looks like it belongs with that statement?

Now look at the actions on offer : Subtract, Divide, Take and Complete. Can you find a line of algebra to go with each of those?

Is there a sequence you can find by following the algebra?