1. ## 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

2. 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).