## Solving the Quadratic Equation: (p+1)x² - 6(p+1)x + 3(p+9) = 0

This article explores the solution of the quadratic equation (p+1)x² - 6(p+1)x + 3(p+9) = 0, where 'p' is a constant. We will use the quadratic formula to find the roots of this equation.

### Understanding the Quadratic Formula

The quadratic formula is a general solution for quadratic equations in the form ax² + bx + c = 0. It is given by:

**x = (-b ± √(b² - 4ac)) / 2a**

Where:

**a**,**b**, and**c**are the coefficients of the quadratic equation.

### Applying the Quadratic Formula to Our Equation

Let's identify the coefficients in our given equation:

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

Now, we can plug these values into the quadratic formula:

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

### Simplifying the Expression

Let's simplify the expression within the square root:

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

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

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

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

### Finding the Roots

Finally, we can simplify further and obtain the two roots of the equation:

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

**x = 3 ± √(3(2p - 6) / (p+1))**

Therefore, the roots of the quadratic equation (p+1)x² - 6(p+1)x + 3(p+9) = 0 are:

**x₁ = 3 + √(3(2p - 6) / (p+1))****x₂ = 3 - √(3(2p - 6) / (p+1))**

These roots are expressed in terms of the constant 'p'. Their values will vary depending on the specific value of 'p'.

### Conclusion

By applying the quadratic formula and simplifying the expression, we have successfully found the roots of the quadratic equation (p+1)x² - 6(p+1)x + 3(p+9) = 0. The roots are dependent on the value of 'p', providing a solution for any given constant 'p'. This process illustrates the power of the quadratic formula in solving quadratic equations with variable coefficients.