(4-k)x^2+(2k+4)x+(8k+1)=0

4 min read Jun 16, 2024

Solving the Quadratic Equation: (4-k)x² + (2k+4)x + (8k+1) = 0

This article will explore the quadratic equation (4-k)x² + (2k+4)x + (8k+1) = 0 and delve into how to find its solutions.

Understanding the Quadratic Formula

The quadratic formula is a fundamental tool for solving equations of the form ax² + bx + c = 0. It states that the solutions for x are given by:

x = (-b ± √(b² - 4ac)) / 2a

Where:

a, b, and c are the coefficients of the quadratic equation.

Applying the Formula to Our Equation

In our equation, (4-k)x² + (2k+4)x + (8k+1) = 0, we have:

a = (4-k)
b = (2k+4)
c = (8k+1)

Substituting these values into the quadratic formula, we get:

x = (-(2k+4) ± √((2k+4)² - 4(4-k)(8k+1))) / 2(4-k)

Simplifying the Expression

Let's simplify the expression under the square root:

(2k+4)² - 4(4-k)(8k+1) = 4k² + 16k + 16 - 128k - 16 + 32k + 4k = 4k² - 92k + 4

Now, our equation becomes:

x = (-(2k+4) ± √(4k² - 92k + 4)) / 2(4-k)

Analyzing the Solutions

The nature of the solutions (real, complex, distinct, or repeated) depends on the discriminant, which is the expression under the square root: 4k² - 92k + 4.

If the discriminant is positive, there are two distinct real solutions.
If the discriminant is zero, there is one repeated real solution.
If the discriminant is negative, there are two complex solutions.

Finding the Solutions for Specific Values of k

To find the solutions for a specific value of k, simply substitute the value into the equation we derived:

x = (-(2k+4) ± √(4k² - 92k + 4)) / 2(4-k)

For example, if k = 1, the equation becomes:

x = (-6 ± √(-84)) / 6 = (-6 ± 2√21i) / 6 = -1 ± (√21i)/3

This demonstrates that for k = 1, there are two complex solutions.

Conclusion

By applying the quadratic formula and simplifying the expression, we have found a general solution for the equation (4-k)x² + (2k+4)x + (8k+1) = 0. The nature of the solutions depends on the discriminant, and specific values of k can be substituted to find the corresponding solutions. This analysis provides a comprehensive understanding of how to solve this quadratic equation and explore its characteristics.