Identifying the Value of k for Equal Roots in Quadratic Equations

When dealing with quadratic equations, understanding the concept of equal roots involves a deep dive into the discriminant. The discriminant, denoted as D, determines the nature of the roots of a quadratic equation in the form of ax^2 bx c 0. If the discriminant equals zero, the equation has equal roots.

Determining the Condition for Equal Roots

To solve the quadratic equation k^3x^2 - 2kx - 1 0 for equal roots, the discriminant D must be zero. The general form of the discriminant for a quadratic equation is D b^2 - 4ac. In our equation:

a k^3 b -2k c -1

Substituting the values, we get:

D (-2k)^2 - 4(k^3)(-1)

D 4k^2 4k^3

D 4k^2(1 k)

Setting D 0, we get:

4k^2(1 k) 0

This equation is zero when either:

4k^2 0 1 k 0

Solving these gives us:

k^2 0 → k 0

1 k 0 → k -1

Exclusion of k -3

It is mentioned in the problem statement that k ≠ -3. This condition must be adhered to when solving the equation. Therefore, the value k -3 does not apply to the problem at hand.

Verification of Equal Roots Condition for Specific Values of k

To ensure the accuracy of the solution, let's verify for specific values of k. We start by substituting k -1 and k -2 into the original equation:

For k -1:

Discriminant 4(-1)^2(1 - 1) 0

For k -2:

Discriminant 4(-2)^2(1 - 2) 0

Both values satisfy the condition for equal roots.

Conclusion

Thus, if the roots of the quadratic equation k^3x^2 - 2kx - 1 0 are equal and k ≠ -3, the values of k that satisfy this condition are k 0 and k -1.

Keywords: quadratic equation, discriminant, equal roots