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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:

$4 x^2 + px + 3 = 0$

#### Solution

The given quadratic equation is  $4 x^2 + px + 3 = 0$ and roots are real and equal.

Then find the value of p.

Here,

$4 x^2 + px + 3 = 0$

So,

$a = 4, b = p \text { and } c = 3 .$

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

Putting the value of

$a = 4, b = p \text { and } c = 3 .$

$D = \left( p \right)^2 - 4\left( 4 \right)\left( 3 \right)$

$= p^2 - 48$

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

So,

$p^2 - 48 = 0$

Now factorizing the above equation,

$p^2 - 48 = 0$

$\Rightarrow p^2 - \left( 4\sqrt{3} \right)^2 = 0$

$\Rightarrow \left( p - 4\sqrt{3} \right)\left( p + 4\sqrt{3} \right) = 0$

$\Rightarrow p - 4\sqrt{3} = 0 \text { or } p + 4\sqrt{3} = 0$

$\Rightarrow p = 4\sqrt{3} \text { or } p = - 4\sqrt{3}$

Therefore, the value of $p = \pm 4\sqrt{3} .$

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