Solving the Quadratic Equation: (k12)x² + 2(k12)x + 2 = 0
This article will guide you through solving the quadratic equation (k12)x² + 2(k12)x + 2 = 0. We'll explore different methods and analyze the conditions for real solutions.
Understanding the Quadratic Formula
The quadratic formula is a powerful tool used to solve any quadratic equation of the form ax² + bx + c = 0. It states that:
x = (b ± √(b²  4ac)) / 2a
where:
 a, b, and c are the coefficients of the quadratic equation.
Applying the Quadratic Formula
Let's apply the quadratic formula to our equation:
 a = (k12)
 b = 2(k12)
 c = 2
Substituting these values into the quadratic formula, we get:
x = (2(k12) ± √((2(k12))²  4(k12)(2))) / 2(k12)
Simplifying the Expression
Let's simplify the expression:
x = (2(k12) ± √(4(k12)²  8(k12))) / 2(k12)
x = (2(k12) ± √(4(k12)(k122))) / 2(k12)
x = (2(k12) ± √(4(k12)(k14))) / 2(k12)
x = (2(k12) ± 2√((k12)(k14))) / 2(k12)
x = ( (k12) ± √((k12)(k14))) / (k12)
Analyzing the Solutions
The solutions to this quadratic equation depend on the value of k:

Case 1: (k12)(k14) > 0
 In this case, the discriminant (the expression under the square root) is positive, resulting in two distinct real solutions.
 The solutions will be real and distinct.

Case 2: (k12)(k14) = 0
 This results in a double root, meaning there is only one solution.
 The solution will be real and repeated.

Case 3: (k12)(k14) < 0
 In this case, the discriminant is negative, resulting in no real solutions.
 The solutions will be complex conjugates.
Conclusion
By applying the quadratic formula and analyzing the discriminant, we've determined the conditions for real solutions to the equation (k12)x² + 2(k12)x + 2 = 0. Remember to consider the different cases based on the value of k to fully understand the nature of the solutions.