Find the value of k for which the equation x² + k(2x + k – 1) + 2 = 0 has real and equal roots.

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Solution

Let x² + k(2x + k – 1) + 2 = 0 be a quadratic equation.

x² + k(2x + k – 1) + 2 = 0
x² + 2xk + k² – k + 2 = 0

It is given that, it has real and equal roots.

$\Rightarrow Discriminant = 0 \Rightarrow b^{2} - 4 a c = 0 \Rightarrow {(2 k)}^{2} - 4 (1) (k^{2} - k + 2) = 0 \Rightarrow 4 k^{2} - 4 k^{2} + 4 k - 8 = 0 \Rightarrow 4 k = 8 \Rightarrow k = 2$

Hence, the value of k is 2.

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