Determining the Value of k for Distinct Roots in Quadratic Equations

Quadratic equations are fundamental in the study of algebra and have various applications in science, engineering, and mathematics. One important aspect of quadratic equations is determining the value of a parameter that ensures the equation has distinct roots. This article explores the process of finding the value of k for which the equation x2 - 8x - 3 k has two distinct roots. This requires an understanding of the discriminant and how it affects the roots of a quadratic equation.

Introduction to Quadratic Equations and Roots

A quadratic equation can be represented in the general form:

ax2 bx c 0

The roots of such an equation can be determined using the quadratic formula:

x (frac{-b pm sqrt{b^2 - 4ac}}{2a})

For a quadratic equation to have distinct roots, its discriminant must be greater than zero.

The Discriminant and Its Significance

The discriminant, denoted as D, is a key factor in determining the nature of the roots. It is calculated as follows:

D b2 - 4ac

If D > 0, the equation has two distinct real roots.

Steps to Determine the Value of k

Step 1: Standard Form of the Equation

The given equation is:

x2 - 8x - 3 k

Let’s rearrange it into standard form:

x2 - 8x - 3 - k 0

Step 2: Identifying the Coefficients

In the standard form, the coefficients are:

a 1 b -8 c -3 - k

Step 3: Calculating the Discriminant

Using the discriminant formula:

D b2 - 4ac

Substituting the identified coefficients:

D (-8)2 - 4(1)(-3 - k)

D 64 12 4k

D 76 4k

Step 4: Ensuring Two Distinct Roots

For the equation to have two distinct roots, the discriminant must be greater than zero:

76 4k > 0

Solving for k:

4k > -76

k > -19

Therefore, for the quadratic equation x2 - 8x - 3 k to have two distinct roots, k must be greater than -19.

Examples and Further Explorations

Let’s consider the equation x2 - 8x - 3 k for different values of k to confirm the result.

Example 1: k -20 (k Example 2: k -18.5 (k > -19) Example 3: k -22 (k Example 4: k -17 (k > -19)

In Examples 1 and 3, the equation will have no real roots or one real root (a repeated root) since the discriminant is not positive. In Examples 2 and 4, the equation will have two distinct real roots since the discriminant is positive.

Conclusion

The value of k for which the quadratic equation x2 - 8x - 3 k has two distinct roots is k > -19. This conclusion is derived from the discriminant formula and the conditions for having distinct roots. Understanding this method is crucial for solving various problems in algebra and provides a clear insight into the behavior of quadratic equations based on the parameter values.