# Equation

• Feb 19th 2011, 08:56 PM
MichaelLight
Equation
Given α and β are the roots of the quadratic equation px^2+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 2q^2 + pq - p^2 = 9pr and 3q^2 + 2pq - p^2 =16pr respectively.

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://img152.imageshack.us/img152/6901/dsc00535l.jpg
• Feb 19th 2011, 11:44 PM
Sudharaka
Quote:

Originally Posted by MichaelLight
Given α and β are the roots of the quadratic equation px^2+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 2q^2 + pq - p^2 = 9pr and 3q^2 + 2pq - p^2 =16pr respectively.

Can you help me?

You should know that if $\displaystyle \alpha$ and $\displaystyle \beta$ are the roots of a quadratic equation, $\displaystyle px^2+qx+r=0$ then the relation between the coefficients and the roots are,

$\displaystyle \alpha+\beta=-\frac{q}{p}$-----(1)

$\displaystyle \alpha\beta=\frac{r}{p}$---------(2)

If $\displaystyle \alpha=2\beta+1$--------(3)

Using equations (1),(2) and (3) eliminate $\displaystyle \alpha~and~\beta$.

Use the same method for part b).