## Solving the Quadratic Equation: (k+1)x^2 - 6(k+1)x + 3(k+9) = 0

This article explores the solution to the quadratic equation (k+1)x^2 - 6(k+1)x + 3(k+9) = 0. We will analyze the equation, determine its roots, and discuss the impact of the parameter 'k'.

### Understanding the Equation

The given equation is a quadratic equation in the variable 'x'. It has the general form ax^2 + bx + c = 0, where:

**a = (k+1)****b = -6(k+1)****c = 3(k+9)**

The value of 'k' is a parameter, meaning it can take on different values, influencing the nature of the quadratic equation.

### Finding the Roots

To solve for the roots of the equation, we can use the quadratic formula:

**x = (-b ± √(b^2 - 4ac)) / 2a**

Substituting the values of a, b, and c:

**x = (6(k+1) ± √((-6(k+1))^2 - 4(k+1)(3(k+9)))) / 2(k+1)**

Simplifying the equation:

**x = (6(k+1) ± √(36(k+1)^2 - 12(k+1)(k+9))) / 2(k+1)**

**x = (6(k+1) ± √(12(k+1)(3(k+1) - (k+9)))) / 2(k+1)**

**x = (6(k+1) ± √(12(k+1)(2k - 6))) / 2(k+1)**

**x = (6(k+1) ± √(24(k+1)(k-3))) / 2(k+1)**

**x = (3(k+1) ± √(6(k+1)(k-3))) / (k+1)**

This gives us two possible roots for the equation:

**x1 = (3(k+1) + √(6(k+1)(k-3))) / (k+1)**

**x2 = (3(k+1) - √(6(k+1)(k-3))) / (k+1)**

### Analyzing the Roots

The roots of the equation are dependent on the value of 'k'.

**If k = -1**, the denominator of both roots becomes zero, resulting in the equation being undefined. This means there are no real roots for k = -1.**If k = 3**, the expression under the square root becomes zero, resulting in two equal roots.**For other values of k**, the equation will have two distinct roots, which can be real or complex depending on the discriminant (the expression under the square root).

### Conclusion

The equation (k+1)x^2 - 6(k+1)x + 3(k+9) = 0 presents a quadratic equation with roots that are influenced by the parameter 'k'. By applying the quadratic formula and analyzing the discriminant, we can determine the nature of the roots and the behavior of the equation for different values of 'k'. Understanding the impact of the parameter on the solution is crucial in analyzing and solving such equations.