Understanding the Problem: Finding k and Roots of a Quadratic Equation

In this article, we will explore a specific problem involving a quadratic equation: Given the equation x^2 - 7x - k - 4 0, we need to determine the value of (k) and the roots if the difference between the roots is 5. We'll go through the steps to solve this problem, providing a clear understanding of the underlying concepts and the mathematical operations involved.

Solve for k

The general form of a quadratic equation is ax^2 bx c 0. For our specific equation, we have:

The discriminant of a quadratic equation is given by D b^2 - 4ac. Substituting the values, we get:

D (-7)^2 - 4 cdot 1 cdot (-k - 4) 49 4k 16 65 4k

The difference of the roots of a quadratic equation can be found using D 65 4k. Given that the difference between the roots is 5, we set up the equation:

sqrt{65 4k} 5

Squaring both sides:

65 4k 25

Solving for (k):

4k -40 rarr; k -10

Therefore, the value of (k) is -10.

Find the Roots of the Quadratic Equation

Now, substitute (k -10) back into the original equation:

x^2 - 7x 6 0

Factoring the quadratic equation:

(x - 1)(x - 6) 0

The roots are:

x 1 and x 6

Summary

To summarize, the value of (k) is -10. The roots of the equation are x 1 and x 6.

Conclusion

This article has elaborated on the step-by-step process to solve a quadratic equation problem involving the difference of roots. The key equations and steps have been clearly demonstrated, making it easier to understand and solve similar problems.