First slide

Theory of equations

Question

If $a \in (- 1, 1)$ then roots of the quadratic equation $(a - 1) x^{2} + ax + \sqrt{1 - a^{2}} = 0$ are

Moderate

A

real
B

imaginary
C

both equal
D

none of these

Solution

$\begin{array}{l} (a - 1) x^{2} + ax + \sqrt{1 - a^{2}} = 0 \\ ∴ D = a^{2} - 4 (a - 1) \sqrt{1 - a^{2}} \\ = a^{2} - 4 a \sqrt{1 - a^{2}} + 4 \sqrt{1 - a^{2}} \\ \begin{array}{clc} {(a - 2 \sqrt{1 - a^{2}})}^{2} + 4 \sqrt{1 - a^{2}} (1 - \sqrt{1 - a^{2}}) & > 0 for a \in (- 1, 1) \end{array} \end{array}$
Hence roots are real but not equal.

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