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Solving the Equation (x-3)^2-2(x-1)=x(x-2)^2-5x^2
This article will guide you through the process of solving the equation (x-3)^2-2(x-1)=x(x-2)^2-5x^2.
1. Expanding the Equation
First, we need to expand the equation by removing the parentheses.
- (x-3)^2 can be expanded as x^2 - 6x + 9
- x(x-2)^2 can be expanded as x(x^2 - 4x + 4) = x^3 - 4x^2 + 4x
Substituting these expansions back into the equation, we get:
x^2 - 6x + 9 - 2x + 2 = x^3 - 4x^2 + 4x - 5x^2
2. Simplifying the Equation
Next, we simplify the equation by combining like terms:
x^2 - 8x + 11 = x^3 - 9x^2 + 4x
3. Rearranging the Equation
Now, we rearrange the equation to have all the terms on one side:
0 = x^3 - 10x^2 + 12x - 11
4. Finding the Solutions
We have a cubic equation. Finding the solutions (the values of 'x' that make the equation true) can be tricky. Here are a few methods:
- Factoring: Try factoring the cubic expression, but this may not always be straightforward.
- Rational Root Theorem: This theorem can help find potential rational roots (roots that are fractions).
- Numerical Methods: Methods like the Newton-Raphson method can approximate solutions.
- Graphing: Plot the function to visually estimate the roots.
For this specific equation, it's likely that finding the exact solutions algebraically will be difficult. You can use numerical methods or graphing tools to approximate the solutions.
5. Conclusion
Solving the equation (x-3)^2-2(x-1)=x(x-2)^2-5x^2 involves expanding, simplifying, and rearranging to get a cubic equation. Finding the solutions may require using numerical methods or graphing tools to approximate the values.