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sakhi telugu movie songs hd 1080p 91Q:

Restrictions for a quadratic function to be have a root in $\mathbb{Q}$

If we have $a,b,c$ positive integers and $a+b+c$ is even and $a^2+b^2+c^2 =abc.$ Prove that $a,b,c$ must all be divisible by 4.
By drawing an equation $ax^2+bx+c = 0$ we can see that it is quadratic and has a root at $\frac{ -b \pm \sqrt{b^2-4ac}}{2a}$. So we want to prove that for a quadratic equation $ax^2+bx+c = 0$ to have a root in $\mathbb{Q}$, we need $b^2-4ac\ge0$.
I have tried to use “root swapping” but it got messy.

A:

Hint:
you know that $a,b,c$ are odd?
Proposition: a quadratic function on $\mathbb Q$ and having the property that both roots are in $\mathbb Q$ has the form
$$
ax^2+bx+c = b^2 – 4ac = (b-2a\sqrt{c/a}) (b + 2a\sqrt{c/a})
$$
Proof:
We have that
$$
a = \frac{b^2-4ac}{4a^2+4bc}
$$
and
$$
c = \frac{(b-2a\sqrt{c/a})(b+2a\sqrt{c/a})}{4a^2+4bc}
$$
$a$ is odd as is $c$, so their product $a^2+c$ is even, and
$$
c = \frac{b^2-4ac}{a^2+c}
$$

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“The Wardrobe” is the fifth episode of the second season of the American prison drama series Orange Is the New Black, which aired on June 3, 2012. The episode was written by Jenji Kohan and directed by David Mackenzie Smith.

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