## Sakhi Telugu Movie Songs Hd 1080p 91

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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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