(x%2B1)%2B(4x%5E3-6x%5E2-6x)-(-2x)%3D18.jpeg "(2x-3)(x+1)+(4x^3-6x^2-6x) (-2x)=18")
Solving the Equation: (2x-3)(x+1)+(4x^3-6x^2-6x) (-2x)=18
This equation involves expanding and simplifying expressions, combining like terms, and solving for the unknown variable, x. Let's break down the steps:
1. Expanding and Simplifying
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Expand the first product: (2x-3)(x+1) = 2x² + 2x - 3x - 3 = 2x² - x - 3
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Expand the second product: (4x³-6x²-6x) (-2x) = -8x⁴ + 12x³ + 12x²
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Substitute the expanded expressions back into the equation: 2x² - x - 3 - 8x⁴ + 12x³ + 12x² = 18
2. Combining Like Terms
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Rearrange the terms in descending order of their exponents: -8x⁴ + 12x³ + 14x² - x - 3 = 18
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Move the constant term to the left side: -8x⁴ + 12x³ + 14x² - x - 21 = 0
3. Solving the Equation
Now we have a polynomial equation. Solving for x in a fourth-degree polynomial equation can be complex and might require advanced techniques such as factoring, the Rational Root Theorem, or numerical methods.
Here's a possible approach using factoring:
- Look for common factors: There are no common factors for all terms.
- Try grouping: This method might work, but it's not always guaranteed.
- Use numerical methods: For more complex equations, numerical methods like the Newton-Raphson method can provide approximations of the solutions.
Important Note: The solution to this equation might involve multiple real or complex roots.
Conclusion
The equation (2x-3)(x+1)+(4x^3-6x^2-6x) (-2x)=18 presents a complex polynomial equation. Finding the solutions for x requires simplifying the equation, combining like terms, and employing appropriate solving techniques. You can use factoring methods or numerical approaches to find the solutions. Remember, there may be multiple solutions to this equation.