Solving Quadratic Equations: Finding the Value of k and Roots Given a Root Difference

In this article, we will explore how to solve for the value of k in a quadratic equation given that the difference of the roots is a specific value. This is a classic problem in algebra and involves applying the properties of quadratic equations to find the roots and the value of the unknown parameter.

Problem Statement

Consider the quadratic equation x2 - 7x - 4 0. We are given that the difference between the roots of this equation is 5. We need to find the value of the parameter k and the roots of the equation.

Solution Steps

Step 1: Using the Difference of Roots Formula

For a quadratic equation of the form ax2 bx c 0, the difference between the roots (denoted as β - α) can be found using the formula:

β - α sqrt{(b2 - 4ac) / a}

Step 2: Applying the Given Information

In our case, the equation is x2 - 7x - 4 0. Here:

a 1 b -7 c -4 k

Given that the difference of the roots is 5, we can set up the equation:

5 sqrt{(-7)2 - 4 × 1 × (-4 k)}

Step 3: Simplifying the Equation

Squaring both sides to eliminate the square root, we get:

25 (-7)2 - 4 × 1 × (-4 k)

25 49 16 - 4k

25 65 - 4k

4k 65 - 25

4k 40

k 10

Step 4: Finding the Roots

Now that we have determined k 10, we substitute it back into the original equation to find the roots:

x2 - 7x - 10 - 4 0

x2 - 7x - 6 0

Step 5: Factoring the Quadratic Equation

Factoring the quadratic equation:

x2 - 7x - 6 0

x - 6)(x 1) 0

The roots of the equation are:

x 6 and x 1

Conclusion

The value of k is 10, and the roots of the equation are 6 and 1.

In summary:

k 10 Roots: 6 and 1

Additional Insights

The steps outlined above involve a combination of algebraic manipulation and the application of the properties of quadratic equations. Understanding these steps can be useful in solving similar problems involving the roots of quadratic equations and the relationships between the coefficients.

For more detailed information on quadratic equations and their properties, you can refer to the following resources:

Math Is Fun: Quadratic Equations Khan Academy: Roots of Quadratic Equations