#### 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 \[p = \pm 4\sqrt{3} .\]