%5E2%2B2(x%2B1%2Fx)-8%3D0.jpeg "(x+1/x)^2+2(x+1/x)-8=0")
Solving the Equation: (x + 1/x)^2 + 2(x + 1/x) - 8 = 0
This equation appears complex at first glance, but we can simplify it using a clever substitution. Let's break down the steps:
1. Substitution:
Let's introduce a new variable, say y, to represent the expression (x + 1/x):
- y = (x + 1/x)
Now our equation becomes much simpler:
- y² + 2y - 8 = 0
2. Solving the Quadratic Equation:
We now have a standard quadratic equation. We can solve it using the quadratic formula:
- y = (-b ± √(b² - 4ac)) / 2a
Where a = 1, b = 2, and c = -8.
-
y = (-2 ± √(2² - 4 * 1 * -8)) / (2 * 1)
-
y = (-2 ± √(36)) / 2
-
y = (-2 ± 6) / 2
This gives us two possible solutions for y:
- y1 = 2
- y2 = -4
3. Finding the Solutions for x:
Now we need to substitute back the original expression for y:
-
For y1 = 2:
- (x + 1/x) = 2
- x² + 1 = 2x
- x² - 2x + 1 = 0
- (x - 1)² = 0
- x = 1
-
For y2 = -4:
- (x + 1/x) = -4
- x² + 1 = -4x
- x² + 4x + 1 = 0
- Using the quadratic formula again, we get:
- x = (-4 ± √(4² - 4 * 1 * 1)) / (2 * 1)
- x = (-4 ± √12) / 2
- x = (-4 ± 2√3) / 2
- x = -2 ± √3
Therefore, the solutions for the equation (x + 1/x)² + 2(x + 1/x) - 8 = 0 are:
- x = 1
- x = -2 + √3
- x = -2 - √3