## Analyzing the Quadratic Equation: (2m+1)x^2 + 2(m+3)x + (m+5) = 0

This article explores the quadratic equation (2m+1)x^2 + 2(m+3)x + (m+5) = 0, examining its properties and potential solutions based on the value of the parameter 'm'.

### Understanding the Equation

The equation is a quadratic equation in the variable 'x'. It is defined by the coefficients (2m+1), 2(m+3), and (m+5) which depend on the parameter 'm'.

### Analyzing the Discriminant

The discriminant of a quadratic equation (ax^2 + bx + c = 0) is given by Δ = b^2 - 4ac. It determines the nature of the roots of the equation.

**For our equation:**

**a = (2m+1)****b = 2(m+3)****c = (m+5)**

Therefore, the discriminant Δ is:

**Δ = [2(m+3)]^2 - 4(2m+1)(m+5)**

**Δ = 4(m^2 + 6m + 9) - 8(m^2 + 7m + 5)**

**Δ = -4m^2 - 20m + 4**

**Δ = -4(m^2 + 5m - 1)**

### Interpreting the Discriminant

The discriminant determines the nature of the roots:

**Δ > 0:**Two distinct real roots.**Δ = 0:**One real root (a repeated root).**Δ < 0:**Two complex roots (conjugate pairs).

To analyze the nature of roots for our equation, we need to examine the expression for Δ:

**-4(m^2 + 5m - 1) = 0**

Solving for 'm' using the quadratic formula or factorization will give us the values of 'm' for which the discriminant is zero. These values will mark the boundaries between different types of roots.

### Determining the Roots

For specific values of 'm', the quadratic formula can be used to find the roots of the equation:

**x = [-b ± √(Δ)] / 2a**

**x = [-2(m+3) ± √(-4(m^2 + 5m - 1))] / 2(2m+1)**

**x = [-(m+3) ± √(-(m^2 + 5m - 1))] / (2m+1)**

This provides the solutions for 'x' based on the value of 'm'.

### Conclusion

The quadratic equation (2m+1)x^2 + 2(m+3)x + (m+5) = 0 exhibits different behaviors based on the value of the parameter 'm'. By analyzing the discriminant and finding the roots, we can fully characterize the equation's solutions and understand its properties. This analysis provides valuable insights into the relationships between parameters and solutions in quadratic equations.