What is the relationship between coefficients and roots in quadratic equations?

[Michael_Light]

Homework Statement

Given α and β are the roots of the quadratic equation px²+qx+r=0, find the relationship between p,q and r if

a) α=2β+1
b) α=3β+1

Answers provided by the answer sheet are 2q² + pq - p² = 9pr and 3q² + 2pq - p² =16pr respectively.

Can anyone help me?

Homework Equations

The Attempt at a Solution

 

[Mark44]

What have you tried? You need to show your attempt before we can provide any help.

 

[Michael_Light]

What have you tried? You need to show your attempt before we can provide any help.

As shown below, i just managed to solve a) but not b), in both solutions for b, i don't know what goes wrong and they are not same as the answer provided, it is possible to be more than one solution? Please help me..

http://img695.imageshack.us/img695/3286/dsc00535o.jpg

 

[eumyang]

I think you started wrong from the very beginning of part b (looking only at the first attempt).

$$\alpha = 3\beta + 1$$
$$\frac{r}{p\beta} = 3\left( \frac{-p-q}{3p}\right) + 1$$

The substitution on the right side was made because of the work you wrote in (2) above, which was based on the work you wrote in (1) above that, which started from the statement
$$\alpha = 2\beta + 1$$
which is the statement from part a, not part b.

You will have to generate new expressions in terms of α and β, starting from
$$\alpha = 3\beta + 1$$
For instance, from here, I get this value for alpha:
$$\alpha = \frac{p-3q}{4p}$$