Question

If 2 is a root of the quadratic equation \[3 x^2 + px - 8 = 0\] and the quadratic equation \[4 x^2 - 2px + k = 0\]  has equal roots, find the value of k.

Answer 1

The given quadratic equation is \[3 x^2 + px - 8 = 0\] and one root is 2.

Then, it satisfies the given equation.

\[3 \left( 2 \right)^2 + p\left( 2 \right) - 8 = 0\]

\[ \Rightarrow 12 + 2p - 8 = 0\]

\[ \Rightarrow 2p = - 4\]

\[ \Rightarrow p = - 2\]

The quadratic equation  \[4 x^2 - 2px + k = 0\],has equal roots.

Putting the value of p, we get

\[4 x^2 - 2( - 2)x + k = 0\]

\[ \Rightarrow 4 x^2 + 4x + k = 0\]

Here, 

\[a = 4, b = 4 \text { and } c = k\].

As we know that

\[D = b^2 - 4ac\]

Putting the values of \[a = 4, b = 4 \text { and } c = k\].

\[D = \left( 4 \right)^2 - 4\left( 4 \right)\left( k \right)\]

\[ = 16 - 16k\]

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

Thus, 

\[16 - 16k = 0\]

\[\Rightarrow 16k = 16\]

\[ \Rightarrow k = 1\]

Therefore, the value of k is 1.