1. ## Roots of a Quadratic equation

Given that $\displaystyle 3p(x^2) - 7qx + 3p = 0$has equal roots
and $\displaystyle p$ and $\displaystyle q$ are positive,
find the ratio $\displaystyle p : q$
and solve the equation.

(I was able to deduce the relation $\displaystyle 49q^2 = 36p$, but how is the ratio further on calculated?
How are the value of $\displaystyle p$ and $\displaystyle q$ deduced in such a way that substituting their values would solve the equation? )

(The answers provided are $\displaystyle 7 : 6$ and $\displaystyle 1$.)

2. $\displaystyle \Delta = (-7q)^2-4\cdot 3p\cdot 3p=49q^2-36p^2$
Because the equation has equal roots $\displaystyle \Delta=0\Rightarrow 49q^2=36p^2\Rightarrow \frac{49}{36}=\frac{p^2}{q^2} \Rightarrow \frac{7}{6}=\frac{p}{q}$

$\displaystyle 3px^2-7qx+3p=0\Leftrightarrow 3\frac{p}{q}x^2-7x+3\frac{p}{q}=0$ Solve it.

3. Originally Posted by Ilsa
Given that $\displaystyle 3p(x^2) - 7qx + 3p = 0$has equal roots
and $\displaystyle p$ and $\displaystyle q$ are positive,
find the ratio $\displaystyle p : q$
and solve the equation.

(I was able to deduce the relation $\displaystyle 49q^2 = 36p$, but how is the ratio further on calculated?
How are the value of $\displaystyle p$ and $\displaystyle q$ deduced in such a way that substituting their values would solve the equation? )

(The answers provided are $\displaystyle 7 : 6$ and $\displaystyle 1$.)
You have a small error. It should be $\displaystyle 49q^2 = 36p^2$ from which it follows that $\displaystyle \frac{p^2}{q^2} = \frac{49}{36}$. Can you take it from here?

4. Another way to do this: if we call the two equal roots "r" then we must have
$\displaystyle 3p(x- r)^2= 3p(x^2- 2rx+ r^2)= 3px^2- 6prx+ 3pr^2= 3px^2- 7qx+ 3p$
and so must have $\displaystyle 6pr= 7q$ and $\displaystyle 3pr^2= 3p$. That second equation reduces to $\displaystyle r^2= 1$ so that r= 1 or r= -1. In order that p and q both be positive, we must have r= 1 and then we have $\displaystyle 6p= 7q$ so that $\displaystyle \frac{p}{q}= \frac{7}{6}$.

5. Hello, Ilsa!

$\displaystyle \text{The answers provided are: }7\!:\!6 \text{ and } 1.$
. . . . . . . . . . . . . . . . . . . . . . . . . . . . $\displaystyle \uparrow$
. . . . . . . . . . . . . . . . . . . . . . . . . . . . $\displaystyle ?$

Are they saying that $\displaystyle p = q$ is a solution?

This is not true . . .

6. "Are they saying that p=q is a solution?" No, they are not saying that...
$\displaystyle \frac{7}{6}$ is $\displaystyle \frac{p}{q}$ and 1 is the solution of that equation.