-(x%2B2%2Fx)-8%3D0.jpeg "(x^2+4/x^2)-(x+2/x)-8=0")
Solving the Equation: (x^2 + 4/x^2) - (x + 2/x) - 8 = 0
This equation might seem intimidating at first, but we can solve it by simplifying it and applying some algebraic techniques. Let's break down the process step-by-step:
1. Simplifying the Equation
- Make a Substitution: Let's simplify the equation by making a substitution. Let y = x + (2/x).
- Rewrite the Equation: Now, we can rewrite the original equation using this substitution: (y^2 - 2) - y - 8 = 0
- Simplify Further: Combine the constants to get: y^2 - y - 10 = 0
2. Solving the Quadratic Equation
We now have a simple quadratic equation. We can solve it using the quadratic formula:
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Quadratic Formula: y = (-b ± √(b^2 - 4ac)) / 2a Where a = 1, b = -1, and c = -10
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Substitute Values: y = (1 ± √((-1)^2 - 4 * 1 * -10)) / 2 * 1
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Calculate: y = (1 ± √41) / 2
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Solutions for y: Therefore, we have two solutions for y:
- y1 = (1 + √41) / 2
- y2 = (1 - √41) / 2
3. Solving for x
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Remember the Substitution: We need to substitute back our original value of y = x + (2/x) to find x.
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Two Equations: We will have two equations to solve:
- x + (2/x) = (1 + √41) / 2
- x + (2/x) = (1 - √41) / 2
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Solve for x: We can solve these equations by multiplying both sides by x, rearranging the terms, and then solving the resulting quadratic equations.
4. Final Solutions
After solving the two equations, we will get four solutions for x. These will be the solutions to the original equation (x^2 + 4/x^2) - (x + 2/x) - 8 = 0.
Important Note: Remember to check your solutions by plugging them back into the original equation to ensure they are valid.