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If x, y and z are real numbers such that: x + y + z = 5 and xy + yz + zx = 3. What is the largest value that any of the numbers can have? In y = ax +b when are a, -b/a, b in arithmetic progression. The polynomial y = ax^2 + bx + c has roots r1 and r2. Can a, r1, b, r2 and c be in arithmetic progression? ### Two Cubes

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# Curve Fitter

##### Age 14 to 18 ShortChallenge Level

This problem challenges you to find cubic equations which satisfy different conditions. You may like to use Desmos to help you investigate possible cubics.

Part 1
Can you find a cubic which passes through $(0,0)$ and the points $(1, 2)$ and $(2,1)$?
Can you find more than one possible cubic?

Can you write down the general algebraic form of a cubic equation?
How can you use this together with the information provided in the question?
Try using this GeoGebra page to investigate possible cubics.

Part 2 (a)
Can you find a cubic which passes through $(0,0)$ and the points $(1, 2)$ and $(2,1)$, and where the point $(1,2)$ is a turning point of the cubic?
Can you find more than one cubic satisfying all the conditions?

What extra information do you now have?

Part 2 (b)
Can you find a cubic which passes through $(0,0)$ and the points $(1, 2)$ and $(2,1)$, and where the point $(2,1)$ is a turning point of the cubic?
Can you find more than one cubic satisfying all the conditions?

Part 3
Can you find a cubic which passes through $(0,0)$ and where the points $(1, 2)$ and $(2,1)$ are both turning points?