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Solving the Equation (x-1)(x-2)(x-4)(x-8) = 7x^2
This equation presents a challenge due to its high degree and the presence of both multiplication and a squared term. Here's a breakdown of how we can approach solving it:
1. Expanding the Equation
The first step is to expand the left side of the equation to get a polynomial form. This involves multiplying out the factors:
(x-1)(x-2)(x-4)(x-8) = 7x^2
Expanding the left side yields:
x⁴ - 15x³ + 74x² - 152x + 128 = 7x²
2. Rearranging and Simplifying
Now, we rearrange the equation to get all terms on one side:
x⁴ - 15x³ + 67x² - 152x + 128 = 0
3. Solving the Polynomial Equation
At this point, we have a quartic equation (an equation with the highest power of x being 4). Solving quartic equations can be quite complex and often requires numerical methods or specialized techniques. Here are a few approaches:
a. Factoring: While factoring directly might be challenging, we can try to find potential rational roots using the Rational Root Theorem. This theorem states that if a polynomial has integer coefficients, any rational root must be of the form p/q, where p is a factor of the constant term (128 in our case) and q is a factor of the leading coefficient (1 in our case). By testing these possible roots, we might be able to factor the polynomial further.
b. Numerical Methods: Numerical methods like the Newton-Raphson method can be used to find approximate solutions for the equation. These methods involve iteratively refining an initial guess until a solution is found within a desired tolerance.
c. Specialized Software: Software like Mathematica or Wolfram Alpha can be used to solve polynomial equations numerically or symbolically.
4. Finding Solutions
It's important to note that a quartic equation can have up to four solutions (real or complex). The specific approach you choose will depend on the tools and techniques available to you.
In conclusion, solving the equation (x-1)(x-2)(x-4)(x-8) = 7x² requires a combination of expansion, rearrangement, and potentially specialized techniques to find the solutions. The process can be challenging due to the nature of the equation, but understanding the methods available can help you find the solutions.