Find positive integers such that the quadratic equation has two real roots both of which are between 0 and 1.

The equation has two real roots, thus Since both roots are between 0 and 1, then then Hence, which implies a unique choice:

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[Solved] Integers quadratic equation

Quadratic Equations

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June 15, 2021 11:36 am

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Find positive integers $m,n$ such that the quadratic equation $4x^2-2mx+n=0$ has two real roots both of which are between 0 and 1.

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June 15, 2021 11:36 am

Topic starter

Staff

(@armor)

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Honorable Member

317 Posts
156 153 0

The equation has two real roots, thus $\small \Delta =4m^2-16n\geq 0\Rightarrow n\leq m^2/4.$

Since both roots are between 0 and 1, then $\small f(0)=n> 0, f(1)=4m-2m+n>0,$ then $\small n>2m-4 (m,n \epsilon \mathbb{N})$

Hence, $\small 2m-4<n\leq m^2/4,$ which implies a unique choice:

$\small m=2, n=1.$

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