First slide

Theory of equations

Question

If the roots of the equation ${ax}^{2} + x + b = 0$ be real, then the roots of the equation $x^{2} - 4 \sqrt{ab} x + 1 = 0$ will be

Moderate

A

Rational
B

Irrational
C

Real
D

Imaginary

Solution

${ax}^{2} + x + b = 0$ has real roots
$\Rightarrow {(1)}^{2} - 4 ab \geq 0 \Rightarrow - 4 ab \geq 1 or 4 ab \leq 1 . . . (i)$
Now second equation is $x^{2} - 4 \sqrt{ab} x + 1 = 0$
Therefore $D = 16 ab - 4$ , from (i) $D \leq 0$
Hence roots are imaginary.

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