Determining the Values of k for Equal Roots in Quadratic Equations

Understanding how to solve quadratic equations where the roots are equal is essential in mathematics. The condition for a quadratic equation to have two equal roots is based on the discriminant, a value that reveals important information about the nature of the roots. This article delves into the process of finding the values of k for which the quadratic equation x2kx160 has two equal roots. We will also explore the general formula and provide examples for a different quadratic equation.

The Importance of the Discriminant

In a quadratic equation of the form ax2 bx c 0, the discriminant D is given by the formula D b2 - 4ac. The discriminant plays a crucial role in determining the nature of the roots:

Positive discriminant: The equation has two distinct real roots. Zero discriminant: The equation has two equal real roots. Negative discriminant: The equation has two complex roots.

Applying the Discriminant to the Given Equation

Consider the quadratic equation x2 kx 16 0. For this equation to have two equal roots, the discriminant must be zero:

Identify the coefficients: a 1, b k, c 16 Calculate the discriminant: D k2 - 4ac k2 - 4 * 1 * 16 k2 - 64 Set the discriminant to zero: k2 - 64 0 Solve for k: k2 64, hence k ±8

Examples and Applications

Let's explore a couple of additional examples to solidify our understanding:

Example 1: x2 kx 16 0

For the equation x2 kx 16 0, we can identify the coefficients and calculate the discriminant as follows:

Identify the coefficients: a 1, b k, c 16 Calculate the discriminant: D k2 - 4ac k2 - 64 Set the discriminant to zero: k2 - 64 0 Solve for k: k2 64, hence k ±8

Example 2: Real and Equal Roots with a 9

Consider another equation: 9x2 kx 16 0. This equation has real and equal roots if the discriminant is zero:

Identify the coefficients: a 9, b k, c 16 Calculate the discriminant: D k2 - 4ac k2 - 4 * 9 * 16 k2 - 576 Set the discriminant to zero: k2 - 576 0 Solve for k: k2 576, hence k ±24

Conclusion

In summary, for a quadratic equation to have two equal roots, the discriminant must be zero. This implies solving the equation k2 - 4ac 0. We have demonstrated this concept through multiple examples, including the specific equation x2 kx 16 0, where k ±8. Understanding these principles is crucial in solving a wide range of mathematical problems.