Determining the Range of Values for k in a Quadratic Equation with Real Roots
When dealing with quadratic equations, one key aspect is determining the conditions under which the roots of the equation are real. This involves analyzing the discriminant, a value derived from the coefficients of the quadratic equation. In this article, we will explore how to determine the range of values for k in the quadratic equation x2 - 2kx 2k2 - 4 0 for which the roots are real.
The Quadratic Equation and its Structure
A quadratic equation generally takes the form ax2 bx c 0. Comparing it to our given equation, we can identify the coefficients as follows:
a 1 b -2k c 2k2 - 4The Role of the Discriminant
The discriminant D of a quadratic equation is given by the formula:
D b2 - 4ac
For the roots to be real, the discriminant must be non-negative (D ≥ 0). We will use this condition to determine the range of values for k.
Calculating the Discriminant
Substituting the values of a, b, and c into the discriminant formula, we get:
D (-2k)2 - 4(1)(2k2 - 4)
Expanding and simplifying this expression:
D 4k2 - 8k2 16
D -4k2 16
For the roots to be real, the discriminant must satisfy:
-4k2 16 ≥ 0
-4k2 ≥ -16
k2 ≤ 4
-2 ≤ k ≤ 2
This means that for the quadratic equation x2 - 2kx 2k2 - 4 0 to have real roots, k must lie within the range [-2, 2].
Conclusion
The range of values for k that guarantees the quadratic equation x2 - 2kx 2k2 - 4 0 has real roots is [-2, 2]. This analysis was carried out by evaluating the discriminant and ensuring it is non-negative.
Additional Notes
Discriminant Analysis: The discriminant of a quadratic equation can provide vital information about the nature of the roots. A positive discriminant indicates two distinct real roots, a zero discriminant indicates exactly one real root (a repeated root), and a negative discriminant indicates two complex conjugate roots.
Application of Quadratic Equations: Understanding when roots are real is crucial in various applications, including physics, engineering, and optimization problems.
Wave-Curve Method and Alternative Solutions: Various methods can be used to solve for the range of k, such as solving using the wave-curve method or algebraic manipulation. The steps provided above can be verified using these alternative methods.