(x+6)(x-2)(x-1)=60

Jasonbradley

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Jun 17, 2024

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Solving the Cubic Equation: (x+6)(x-2)(x-1) = 60

This article explores the solution to the cubic equation (x+6)(x-2)(x-1) = 60. We'll break down the steps to find the values of 'x' that satisfy the equation.

Expanding and Simplifying

1. Expand the left-hand side: Begin by multiplying the factors together.

   • (x+6)(x-2) = x² + 4x - 12
   • (x² + 4x - 12)(x-1) = x³ + 3x² - 16x + 12

2. Rearrange the equation: Move the constant term to the left side to set the equation to zero.

   • x³ + 3x² - 16x + 12 - 60 = 0
   • x³ + 3x² - 16x - 48 = 0

Finding the Solutions

Now we have a cubic equation in standard form. There are a few ways to find the solutions (roots) of this equation:

1. Rational Root Theorem:

• This theorem helps us find potential rational roots. It states that if a polynomial equation has integer coefficients, then any rational root must be of the form p/q, where p is a factor of the constant term (-48) and q is a factor of the leading coefficient (1).
• Factors of -48: ±1, ±2, ±3, ±4, ±6, ±8, ±12, ±16, ±24, ±48
• Factors of 1: ±1
• Test these possible rational roots using synthetic division or direct substitution.

2. Factoring:

• Look for patterns and try to factor the cubic expression. In this case, we can factor by grouping:
  • x³ + 3x² - 16x - 48 = x²(x + 3) - 16(x + 3)
  • (x² - 16)(x + 3) = 0
  • (x + 4)(x - 4)(x + 3) = 0
• This gives us the solutions: x = -4, x = 4, and x = -3.

3. Numerical Methods:

• If factoring or the Rational Root Theorem don't yield solutions, numerical methods like the Newton-Raphson method can be used to approximate the roots.

Conclusion

The solutions to the equation (x+6)(x-2)(x-1) = 60 are:

• x = -4
• x = 4
• x = -3

These values satisfy the original equation when substituted back.