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(x+1)(x-1)(x-2)(x-4)=7

4 min read Jun 16, 2024
(x+1)(x-1)(x-2)(x-4)=7

Solving the Equation (x+1)(x-1)(x-2)(x-4) = 7

This equation involves a product of four factors set equal to a constant. Let's break down the steps to solve for the values of x.

1. Expanding the Equation

First, we need to expand the left side of the equation to get a polynomial. We can do this by using the distributive property (or FOIL method) multiple times:

  • Step 1: Expand (x+1)(x-1) which results in (x² - 1)
  • Step 2: Expand (x-2)(x-4) which results in (x² - 6x + 8)

Now our equation becomes: (x² - 1)(x² - 6x + 8) = 7

  • Step 3: Expand the entire left side using the distributive property again:

    x⁴ - 6x³ + 8x² - x² + 6x - 8 = 7

  • Step 4: Combine like terms:

    x⁴ - 6x³ + 7x² + 6x - 15 = 0

2. Finding the Solutions

We now have a fourth-degree polynomial equation. Unfortunately, there's no general formula to solve equations of this degree directly. Here are a couple of approaches:

  • Factoring: Try to factor the polynomial. This might be challenging for a fourth-degree equation.
  • Numerical Methods: Use numerical methods like the Newton-Raphson method or graphing calculators to find approximate solutions.

3. Approximating Solutions (using Numerical Methods)

Since factoring might be difficult, let's consider using a numerical method like the Newton-Raphson method. This method iteratively refines an initial guess to find a root (a solution where the equation equals zero).

Important Note: The Newton-Raphson method can be complex to implement by hand. It's often easier to use a graphing calculator or online tools that have this functionality built in.

Here are the steps to find a solution using a graphing calculator or similar tool:

  • Graph the Function: Graph the function y = x⁴ - 6x³ + 7x² + 6x - 15
  • Identify X-Intercepts: The x-intercepts of the graph represent the solutions to the equation.
  • Use "Solve" or "Zero" Function: Many calculators have a function to find the x-intercept of a graph. This will provide the numerical approximation of the solution.

By using these methods, you'll find that the equation has four real solutions (values of x that satisfy the equation).

Remember: Numerical methods often provide approximations, especially for higher-degree polynomials.

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