First slide

Theory of equations

Question

If $k \in (- \infty, - 2) \cup (2, \infty),$ then the roots of the equation $x^{2} + 2 kx + 4 = 0$ are

Moderate

A

Complex
B

Real and unequal
C

Real and equal
D

One real and one imaginary

Solution

Given equation is $x^{2} + 2 kx + 4 = 0$
Put $k = - 3, x^{2} - 6 x + 4 = 0 \Rightarrow x = 3 + \sqrt{5}, 3 - \sqrt{5}$
Put $k = 3, x^{2} + 6 x + 4 = 0 \Rightarrow x = - 3 + \sqrt{5}, - 3 - \sqrt{5}$
i.e., Roots are real and unequal.

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