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# Find the Value of K for Which the Following Equations Have Real and Equal Roots: - Mathematics

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#### Question

Find the value of k for which the following equations have real and equal roots:

$x^2 + k\left( 2x + k - 1 \right) + 2 = 0$

#### Solution

The given equation is  $x^2 + k\left( 2x + k - 1 \right) + 2 = 0$

$\Rightarrow x^2 + 2kx + k\left( k - 1 \right) + 2 = 0$

So, a = 1, b = 2k, c = k(k − 1) + 2
We know

$D = b^2 - 4ac$

$\Rightarrow D = \left( 2k \right)^2 - 4 \times 1 \times \left[ k\left( k - 1 \right) + 2 \right]$

$\Rightarrow D = 4 k^2 - 4\left[ k^2 - k + 2 \right]$

$\Rightarrow D = 4 k^2 - 4 k^2 + 4k - 8$

$\Rightarrow D = 4k - 8 = 4\left( k - 2 \right)$

For equal roots, D = 0
Thus, 4(k − 2) = 0
So, k = 2.

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