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# Find the Values of K for Which the Roots Are Real and Equal in Each of the Following Equation: - Mathematics

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#### Question

Find the values of k for which the roots are real and equal in each of the following equation:

$kx\left( x - 2\sqrt{5} \right) + 10 = 0$

#### Solution

The given quadratic equation is  $kx\left( x - 2\sqrt{5} \right) + 10 = 0$ and roots are real and equal.

Then find the value of k.

Here,

$kx\left( x - 2\sqrt{5} \right) + 10 = 0$

$\Rightarrow k x^2 - 2\sqrt{5}kx + 10 = 0$

So,$a = k, b = - 2\sqrt{5}k \text { and c }= 10 .$

As we know that $D = b^2 - 4ac$

Putting the value of

$a = k, b = - 2\sqrt{5}k \text { and } c = 10 .$

$D = \left( - 2\sqrt{5}k \right)^2 - 4\left( k \right)\left( 10 \right)$

$= 20 k^2 - 40k$

The given equation will have real and equal roots, if D = 0.

So, $20 k^2 - 40k = 0$

Now factorizing the above equation,

$20 k^2 - 40k = 0$

$\Rightarrow 20k\left( k - 2 \right) = 0$

$\Rightarrow 20k = 0 \text { or } k - 2 = 0$

$\Rightarrow k = 0 \text { or } k = 2$

Therefore, the value of  $k = 0, 2$.

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