First slide

Theory of equations

Question

If the roots of the equation $x^{2} + 2 ax + b = 0$ are real and distinct and they differ by at most 2m, then b lies in the interval

Moderate

A

$(a^{2}, a^{2} + m^{2})$
B

$(a^{2} - m^{2}, a^{2})$
C

$[a^{2} - m^{2}, a^{2})$
D

none of these

Solution

Let the roots be $α, β$
$∴ α + β = - 2 a and αβ = b$
Given, $| α - β | \leq 2 m$
or $| α - β |^{2} \leq (2 m)^{2}$
or $(α + β)^{2} - 4 ab \leq 4 m^{2}$
or $4 a^{2} - 4 b \leq 4 m^{2}$
$\Rightarrow a^{2} - m^{2} \leq b$ and discriminant $D > 0 or 4 a^{2} - 4 b > 0$
$\Rightarrow a^{2} - m^{2} \leq b and b < a^{2}$
Hence, $b \in [a^{2} - m^{2}, a^{2})$

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