-x2%2B2(k-12)-x%2B2%3D0.jpeg "(k-12) X2+2(k-12) X+2=0")
Solving the Quadratic Equation: x² + 2(k-12)x + 2 = 0
This article will explore the quadratic equation x² + 2(k-12)x + 2 = 0, where 'k' is a constant. We'll analyze its properties, discuss different methods to solve it, and look at how to find solutions for different values of 'k'.
Understanding the Equation
The given equation is a quadratic equation, meaning it is a polynomial equation with the highest power of 'x' being 2. It is in the standard form:
ax² + bx + c = 0
Where:
- a = 1
- b = 2(k-12)
- c = 2
The solutions to a quadratic equation are also known as its roots or zeros. They represent the values of 'x' that make the equation true.
Methods to Solve the Quadratic Equation
There are several methods to solve quadratic equations. Here are two common approaches:
-
Quadratic Formula:
The quadratic formula is a general solution for any quadratic equation. It is derived from completing the square and provides the roots of the equation:
x = (-b ± √(b² - 4ac)) / 2a
Substituting the values from our equation:
x = (-2(k-12) ± √((2(k-12))² - 4 * 1 * 2)) / (2 * 1)
x = (-2(k-12) ± √(4(k-12)² - 8)) / 2
x = -(k-12) ± √((k-12)² - 2)
-
Factoring:
Factoring involves finding two binomials that multiply to give the original quadratic expression. However, this method isn't always applicable and may not be the most efficient approach in this case.
Finding Solutions for Different Values of 'k'
The nature of the solutions (roots) to the quadratic equation depends on the discriminant, which is the expression under the radical in the quadratic formula: b² - 4ac.
- Discriminant > 0: The equation has two distinct real roots.
- Discriminant = 0: The equation has one real root (a double root).
- Discriminant < 0: The equation has two complex roots.
In our case, the discriminant is (k-12)² - 2. Let's analyze the different scenarios:
- k = 12 ± √2: The discriminant is zero, resulting in one real root.
- k > 12 + √2 or k < 12 - √2: The discriminant is positive, leading to two distinct real roots.
- 12 - √2 < k < 12 + √2: The discriminant is negative, leading to two complex roots.
Conclusion
By understanding the quadratic formula and the discriminant, we can determine the nature and calculate the solutions to the equation x² + 2(k-12)x + 2 = 0 for any value of 'k'. This analysis provides valuable insights into the behavior of quadratic equations and their applications in various fields, including mathematics, physics, and engineering.