(x2%2B2x%2B7)%2B2(x2-4)-5(x-2)%3D0.jpeg "(x-2)(x2+2x+7)+2(x2-4)-5(x-2)=0")
Solving the Equation: (x-2)(x^2+2x+7) + 2(x^2-4) - 5(x-2) = 0
This equation looks a bit intimidating at first glance, but we can solve it systematically using a few algebraic techniques. Let's break it down step-by-step:
1. Expanding the Equation
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Distribute: Begin by expanding the products in the equation:
- (x-2)(x^2+2x+7) = x^3 + 2x^2 + 7x - 2x^2 - 4x - 14 = x^3 + 3x - 14
- 2(x^2-4) = 2x^2 - 8
- -5(x-2) = -5x + 10
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Combine terms: Now, rewrite the entire equation with the expanded terms: x^3 + 3x - 14 + 2x^2 - 8 - 5x + 10 = 0
2. Simplifying the Equation
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Combine like terms: Group the terms with the same powers of x: x^3 + 2x^2 + (3x - 5x) + (-14 - 8 + 10) = 0
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Simplify: Combine the coefficients: x^3 + 2x^2 - 2x - 12 = 0
3. Finding the Solutions
At this point, we have a cubic equation. Solving cubic equations can be quite complex, and there's no single, straightforward formula like the quadratic formula. Here are some possible approaches:
- Factoring: Try to factor the equation. In this case, it might be tricky to factor directly.
- Rational Root Theorem: This theorem helps you find potential rational roots (roots that can be expressed as fractions). If you find a rational root, you can divide the equation by (x - root) to reduce it to a quadratic equation.
- Numerical Methods: Methods like Newton-Raphson iteration can approximate the solutions of the equation.
It's important to note that without further information or specific instructions, finding the exact solutions to this cubic equation might require more advanced techniques and tools.
Key Points
- Simplification: Always simplify the equation as much as possible before attempting to solve it.
- Techniques: Be familiar with different techniques for solving polynomials, especially cubic equations.
- Approximation: If finding exact solutions is too difficult, consider using numerical methods to approximate them.