## Analyzing the Quadratic Equation: (m+1)x² + 2(m+3)x + (m+8) = 0

This article explores the quadratic equation (m+1)x² + 2(m+3)x + (m+8) = 0, focusing on its properties and how to solve for its roots.

### Understanding the Equation

The given equation is a quadratic equation in the variable *x*. It is of the general form ax² + bx + c = 0, where:

**a = m + 1:**The coefficient of the quadratic term.**b = 2(m + 3):**The coefficient of the linear term.**c = m + 8:**The constant term.

The values of 'a', 'b', and 'c' are dependent on the parameter 'm'. This means the behavior and roots of the equation can vary depending on the value of 'm'.

### Solving for the Roots

We can solve for the roots of the quadratic equation using the quadratic formula:

**x = [-b ± √(b² - 4ac)] / 2a**

Substituting the values of 'a', 'b', and 'c' from our equation, we get:

**x = [-2(m+3) ± √(4(m+3)² - 4(m+1)(m+8))] / 2(m+1)**

Simplifying the expression:

**x = [-(m+3) ± √((m+3)² - (m+1)(m+8))] / (m+1)**

**x = [-(m+3) ± √(m² + 6m + 9 - m² - 9m - 8)] / (m+1)**

**x = [-(m+3) ± √(-3m + 1)] / (m+1)**

This gives us two possible solutions for *x*, depending on the value of 'm'.

### Analyzing the Roots

The nature of the roots depends on the discriminant, which is the expression under the square root in the quadratic formula:

**Δ = b² - 4ac = -3m + 1**

**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.

To determine the nature of the roots for specific values of 'm', we need to analyze the discriminant.

For example:

**If m = 0:**Δ = 1, indicating two distinct real roots.**If m = 1/3:**Δ = 0, indicating one real root.**If m = 1:**Δ = -2, indicating two complex roots.

### Conclusion

The quadratic equation (m+1)x² + 2(m+3)x + (m+8) = 0 exhibits diverse behaviors depending on the parameter 'm'. Its roots can be real, complex, or even coincident, making it an interesting case study for understanding the dynamics of quadratic equations. By analyzing the discriminant and solving for the roots using the quadratic formula, we can gain deeper insights into the behavior of this equation for various values of 'm'.