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Solving the Quadratic Equation: (3m+1)x² + 2(m+1)x + m = 0
This article will explore the quadratic equation (3m+1)x² + 2(m+1)x + m = 0, examining its solutions and the conditions under which they exist.
Understanding Quadratic Equations
A quadratic equation is a polynomial equation of the second degree, meaning the highest power of the variable is 2. The general form of a quadratic equation is:
ax² + bx + c = 0
where a, b, and c are constants and a ≠ 0.
Solving the Quadratic Equation
To solve for x in our specific equation (3m+1)x² + 2(m+1)x + m = 0, we can employ the quadratic formula:
x = (-b ± √(b² - 4ac)) / 2a
In our equation:
- a = 3m + 1
- b = 2(m + 1)
- c = m
Substituting these values into the quadratic formula, we get:
x = (-2(m + 1) ± √((2(m + 1))² - 4(3m + 1)(m))) / 2(3m + 1)
Analyzing the Solutions
The nature of the solutions (real or complex) depends on the discriminant, the expression under the square root:
Δ = b² - 4ac
- Δ > 0: The equation has two distinct real solutions.
- Δ = 0: The equation has one real solution (a double root).
- Δ < 0: The equation has two complex solutions.
In our case, the discriminant is:
Δ = (2(m + 1))² - 4(3m + 1)(m)
Simplifying this expression:
Δ = 4m² + 8m + 4 - 12m² - 4m = -8m² + 4m + 4
To analyze the solutions, we need to consider the value of the discriminant for different values of m.
Conclusion
This exploration of the quadratic equation (3m+1)x² + 2(m+1)x + m = 0 has revealed the importance of the quadratic formula and the discriminant in determining the nature and number of solutions. By analyzing the discriminant, we can understand whether the equation has real or complex solutions. Further analysis of the discriminant and its relationship with the value of m would provide deeper insights into the behavior of this equation.