Q. For how many values of $k,\ x^{2} + x + 1 + 2k\ \left( x^{2} - x - 1 \right) = 0$ is a perfect square?
A) 2
B) 0
C) 1
D) 3
Explanation
(a)
Given, $x^{2}(1 + 2k) + x(1 - 2k) + 1(1 - 2k) = 0$ ...(i)
Given, $D = 0,\ \ \ b^{2} - 4ac = 0$
$\Rightarrow \ \ (1 - 2k)^{2} - 4(1 + 2k)(1 - 2k) = 0$
$\Rightarrow 20k^{2} - 4k - 3 = 0$
$\Rightarrow k = \frac{1}{2},\frac{3}{10}$
Reviewed by Dr. Abubacker
Physics Educator & NEET/JEE Specialist
Ph.D. in Physics with 19+ years of teaching experience.