x2%2B2(k%2B3)x%2B(k%2B5)%3D0.jpeg "(2k+1)x2+2(k+3)x+(k+5)=0")
Analyzing the Quadratic Equation: (2k+1)x² + 2(k+3)x + (k+5) = 0
This article explores the quadratic equation (2k+1)x² + 2(k+3)x + (k+5) = 0, focusing on its properties, solutions, and the impact of the parameter 'k'.
Understanding the Equation
This is a quadratic equation in the variable 'x', with coefficients involving the parameter 'k'. The general form of a quadratic equation is ax² + bx + c = 0. In this case:
- a = (2k+1)
- b = 2(k+3)
- c = (k+5)
Finding the Solutions
To solve for 'x', we can utilize the quadratic formula:
x = (-b ± √(b² - 4ac)) / 2a
Substituting the coefficients:
x = (-2(k+3) ± √((2(k+3))² - 4(2k+1)(k+5))) / 2(2k+1)
Simplifying the expression:
x = (-2k - 6 ± √(4k² + 24k + 36 - 8k² - 48k - 20)) / (4k + 2)
x = (-2k - 6 ± √(-4k² - 24k + 16)) / (4k + 2)
x = (-k - 3 ± √(-k² - 6k + 4)) / (2k + 1)
This gives us two solutions for 'x', depending on the value of 'k'.
The Role of the Parameter 'k'
The parameter 'k' significantly impacts the nature of the solutions. Let's analyze:
- Discriminant: The expression under the square root, (-k² - 6k + 4), is known as the discriminant. It determines the nature of the solutions:
- If the discriminant is positive ( > 0): The equation has two distinct real roots.
- If the discriminant is zero ( = 0): The equation has one real root (a double root).
- If the discriminant is negative ( < 0): The equation has two complex roots (conjugate pairs).
- Coefficients: The values of 'k' also influence the coefficients 'a', 'b', and 'c', affecting the shape and position of the parabola representing the quadratic equation.
Example: Finding Solutions for a Specific 'k'
Let's say k = 1. Substituting into the solution formula:
x = (-1 - 3 ± √(-1² - 6(1) + 4)) / (2(1) + 1)
x = (-4 ± √(-3)) / 3
x = (-4 ± i√3) / 3
This shows that for k = 1, the quadratic equation has two complex roots.
Conclusion
The equation (2k+1)x² + 2(k+3)x + (k+5) = 0 presents a rich mathematical exploration. By analyzing the discriminant and the influence of the parameter 'k', we can understand the nature and behavior of its solutions. This knowledge is essential for solving quadratic equations and for further applications in fields like calculus and physics.