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

This article will delve into the solution of the quadratic equation (k+1)x² + 2(k+3)x + (k+8) = 0, exploring its roots and the conditions under which these roots are real, distinct, or equal.

### Understanding the Problem

The given equation is a quadratic equation in the variable 'x'. It's important to understand that the value of 'k' determines the nature of the roots. To find the roots, we can utilize the quadratic formula.

### Quadratic Formula

The quadratic formula is a standard method to solve equations of the form ax² + bx + c = 0. It states that:

**x = [-b ± √(b² - 4ac)] / 2a**

Where:

**a**is the coefficient of x²**b**is the coefficient of x**c**is the constant term

### Applying the Quadratic Formula

In our equation, we have:

- a = (k+1)
- b = 2(k+3)
- c = (k+8)

Substituting these values into the quadratic formula, we get:

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

### Analyzing the Discriminant

The term under the square root, (b² - 4ac), is called the **discriminant (Δ)**. It determines the nature of the roots of the quadratic equation.

**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 (not real).

Let's calculate the discriminant for our equation:

**Δ = 4(k+3)² - 4(k+1)(k+8)**

Simplifying, we get:

**Δ = 4(k² + 6k + 9) - 4(k² + 9k + 8)**

**Δ = -12k + 4**

### Conditions for Different Roots

Now, we can analyze the discriminant to determine the conditions for different types of roots:

**1. Two Distinct Real Roots:**

For the equation to have two distinct real roots, the discriminant must be greater than zero:

**Δ > 0**
**-12k + 4 > 0**
**-12k > -4**
**k < 1/3**

**2. One Real Root (Double Root):**

For the equation to have one real root, the discriminant must be equal to zero:

**Δ = 0**
**-12k + 4 = 0**
**-12k = -4**
**k = 1/3**

**3. Two Complex Roots:**

For the equation to have two complex roots, the discriminant must be less than zero:

**Δ < 0**
**-12k + 4 < 0**
**-12k < -4**
**k > 1/3**

### Conclusion

Therefore, the solution of the quadratic equation (k+1)x² + 2(k+3)x + (k+8) = 0 depends on the value of 'k'. We have established the following conditions:

**k < 1/3:**The equation has two distinct real roots.**k = 1/3:**The equation has one real root (a double root).**k > 1/3:**The equation has two complex roots.