Find the value of $k$ for which the following equation has equal roots -
$x^{2} + 4 k x + (k^{2} - k + 2) = 0$

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Solution

Step 1-Compare the equation with the general equation $a x^{2} + b x + c = 0$ :
$x^{2} + 4 k x + (k^{2} - k + 2) = 0$
compare it with the general equation $a x^{2} + b x + c = 0$
$a = 1, b = 4 k, c = k^{2} - k + 2$
Step 2-Find the value of $k$ :
for the roots to be equal discriminant must be equal to $0$
$D = b^{2} - 4 a c 0 = {(4 k)}^{2} - 4 \times (1) \times (k^{2} - k + 2) 0 = 16 k^{2} - 4 k^{2} + 4 k - 8 12 k^{2} + 4 k - 8 = 0 3 k^{2} + k - 2 = 0 3 k^{2} + 3 k - 2 k - 2 = 0 3 k (k + 1) - 2 (k + 1) = 0 (3 k - 2) (k + 1) = 0 k = \frac{2}{3} a n d k = - 1$
Hence the value of $k$ in the equation $x^{2} + 4 k x + (k^{2} - k + 2) = 0$ having equal roots is $k = - 1$ .or $k = \frac{2}{3}$

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