Find the exact values of k for which kx^2- kx - 2x - 3 = 0 has equal roots

Answer 1

To find the exact values of k for which the equation has equal roots, we can use the discriminant.

The discriminant of a quadratic equation, ax^2 + bx + c = 0, is given by b^2 - 4ac. If the discriminant is equal to zero, the quadratic equation has equal roots.

In the given equation, kx^2 - kx - 2x - 3 = 0, the quadratic term coefficient is k, the linear term coefficient is (-k - 2), and the constant term is -3.

Substituting these values into the discriminant formula, we get:
(-k - 2)^2 - 4(k)(-3)

Expanding the equation gives us:
k^2 + 4k + 4 - 4k + 24

Simplifying further:
k^2 + 24 + 4

Combining like terms, we have:
k^2 + 28

Since we want the equation to have equal roots, the discriminant should be equal to zero.

Setting the discriminant equal to zero, we have:
k^2 + 28 = 0

Rearranging the equation:
k^2 = -28

Taking the square root of both sides, we get:
k = ±√(-28)

Since the square root of a negative number is not a real number, there are no real values of k for which the equation has equal roots.