Find the Values of k for Which the Given Quadratic Equation Has Real and Distinct Roots

Question:

\( kx^2 + 6x + 1 = 0 \)

Find the values of \(k\) for which the above quadratic equation has real and distinct roots.

Solution

For real and distinct roots, the discriminant must be positive.

\( D = b^2 – 4ac > 0 \)

Here,

\( a = k,\quad b = 6,\quad c = 1 \)

Substituting these values,

\( 6^2 – 4(k)(1) > 0 \)

\( 36 – 4k > 0 \)

\( 9 – k > 0 \)

\( k < 9 \)

Since the equation is quadratic,

\( k \ne 0 \)

Answer

\( k < 9,\; k \ne 0 \)