If 2 is a Root of 3x² + px − 8 = 0 and 4x² − 2px + k = 0 Has Equal Roots, Find k

Question:

If \(2\) is a root of the quadratic equation

\(3x^2+px-8=0\)

and the quadratic equation

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

has equal roots, find the value of \(k\).

Solution

Since \(2\) is a root of the first equation,

\(3(2)^2+2p-8=0\)

\(12+2p-8=0\)

\(2p+4=0\)

\(p=-2\)

Substituting \(p=-2\) in the second equation,

\(4x^2+4x+k=0\)

For equal roots, the discriminant must be zero.

\(D=b^2-4ac=0\)

\(4^2-4(4)(k)=0\)

\(16-16k=0\)

\(k=1\)

Answer

\( \boxed{k=1} \)