(p+1)x^2-6(p+1)x+3(p+9)=0

4 min read Jun 16, 2024

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.