If 3x2+2αxy+2y2+2ax-4y+1 can be resolved into two linear factors, prove that α is the root of the equation x2+ 4ax+2a2+6=0.

please don't solve the problem

just hint is expected.

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- Aug 16th 2011, 07:13 PMsumedhproblem of quadratic equation with two variables
If 3x2+2αxy+2y2+2ax-4y+1 can be resolved into two linear factors, prove that α is the root of the equation x2+ 4ax+2a2+6=0.

please don't solve the problem

just hint is expected. - Aug 16th 2011, 11:34 PMFernandoRevillaRe: problem of quadratic equation with two variables
- Aug 17th 2011, 03:26 AMHallsofIvyRe: problem of quadratic equation with two variables
That is, if you meant "can be resolved into two linear factors"

**with rational coefficients**. - Aug 17th 2011, 04:28 AMFernandoRevillaRe: problem of quadratic equation with two variables
Why?. The problem is equivalent to find where the given conic is degenerated. That is $\displaystyle \Delta=\begin{vmatrix}{3}&{\alpha}&{a}\\{\alpha}&{ 2}&{-2}\\{a}&{-2}&{1}\end{vmatrix}=0$ i.e. $\displaystyle \alpha^2+4a\alpha +2a^2+6=0$ . Why do we need rational coefficients ?

- Aug 17th 2011, 08:37 AMsumedhRe: problem of quadratic equation with two variables
on solving i got

α^2 y^2+a^2+2aαy =-6y^2-3+12y [α means alpha]

α^2 y^2+a^2+2aαy -9 (y-3)^2

after equating what should i do???

could you please tell me the concept for solving this???? - Aug 17th 2011, 09:29 AMFernandoRevillaRe: problem of quadratic equation with two variables
The discriminant of the equation $\displaystyle p(x,y)=3x^2+(2\alpha y+2a)x+2y^2-4y+1=0$ is $\displaystyle \Delta=(2\alpha y+2a)^2-12(2y^2-4y+1)$ then,

$\displaystyle p(x,y)=3\left(x-\dfrac{-(2\alpha y+a)+\sqrt{\Delta}}{6}\right)\left(x-\dfrac{-(2\alpha y+a)-\sqrt{\Delta}}{6}\right)$

$\displaystyle \Delta$ is a perfect square iff the discriminant $\displaystyle \Delta_1$ of $\displaystyle \Delta=0$ is $\displaystyle 0$ . Now, verify $\displaystyle \Delta_1=0\Leftrightarrow \ldots \Leftrightarrow \alpha^2+4\alpha a+2a^2+6=0$ . Equivalently, $\displaystyle \alpha$ is a root of $\displaystyle x^2+4ax+2a^2+6=0$ . - Aug 19th 2011, 11:49 PMsumedhRe: problem of quadratic equation with two variables
thank you very much i got it