First slide

Theory of equations

Question

$If the roots of the equation x^{2} - a x + b = 0 are real and differ by a quantity which is less than c (c > 0), then b lies between$

Moderate

A

$\frac{a^{2} - c^{2}}{4} and \frac{a^{2}}{4}$
B

$\frac{a^{2} + c^{2}}{4} and \frac{a^{2}}{4}$
C

$\frac{a^{2} - c^{2}}{2} and \frac{a^{2}}{4}$
D

None of these

Solution

Given roots are real and distinct,
$then a^{2} - 4 b > 0 \Rightarrow b < a^{2} / 4 - - - - (1)$
$Again α and β differ by a quantity less than c (c > 0)$
$\begin{array}{l} \Rightarrow | α - β | < c or (α - β)^{2} < c^{2} \\ \Rightarrow (α + β)^{2} - 4 α β < c^{2} \end{array}$
$or a^{2} - 4 b < c^{2} or \frac{a^{2} - c^{2}}{4} < b - - - - - (2)$
$\Rightarrow \frac{a^{2} - c^{2}}{4} < b < \frac{a^{2}}{4} by (1) and (2)$

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