x%5E2-2(m%2B4)x%2B5m%2B2%3D0.jpeg "(2m-1)x^2-2(m+4)x+5m+2=0")
Analyzing the Quadratic Equation: (2m-1)x^2 - 2(m+4)x + 5m + 2 = 0
This article will delve into the quadratic equation (2m-1)x^2 - 2(m+4)x + 5m + 2 = 0, exploring its characteristics and analyzing its solutions.
Understanding the Equation
This equation is a quadratic equation in the variable x. The coefficients of the equation are functions of the parameter m. This means the nature of the solutions (roots) of the equation will depend on the value of m.
Discriminant and Nature of Roots
The discriminant of a quadratic equation (ax^2 + bx + c = 0) is given by Δ = b^2 - 4ac. This value helps determine the nature of the roots:
- Δ > 0: The equation has two distinct real roots.
- Δ = 0: The equation has one real root (a double root).
- Δ < 0: The equation has two complex roots (conjugate pairs).
For the equation (2m-1)x^2 - 2(m+4)x + 5m + 2 = 0, the discriminant is:
Δ = (-2(m+4))^2 - 4(2m-1)(5m+2)
Simplifying, we get:
Δ = -36m^2 - 80m + 80
We can now analyze the nature of the roots by studying the discriminant:
- Δ > 0: The equation will have two distinct real roots if -36m^2 - 80m + 80 > 0. This inequality can be solved to find the range of values for m where this condition is true.
- Δ = 0: The equation will have one real root if -36m^2 - 80m + 80 = 0. Solving this equation will give the values of m for which there is a double root.
- Δ < 0: The equation will have two complex roots if -36m^2 - 80m + 80 < 0. Solving this inequality will find the range of values for m where the roots are complex.
Finding the Roots
The roots of the quadratic equation can be found using the quadratic formula:
x = (-b ± √Δ) / 2a
where a = (2m-1), b = -2(m+4), and c = 5m + 2.
Substituting the values of a, b, c, and Δ, we can find the roots for different values of m.
Conclusion
By understanding the discriminant and the quadratic formula, we can fully analyze the quadratic equation (2m-1)x^2 - 2(m+4)x + 5m + 2 = 0. We can determine the nature of its roots and find their exact values for different values of the parameter m. This analysis provides a deeper understanding of how the equation behaves and its relationship to the parameter m.