Understanding the Problem
To solve for the value of (x² - 1) given that x² + 1/(x²) = 27, we start by manipulating the equation.
Step 1: Rearranging the Equation
We can express 1/(x²) in terms of x²:
- Let y = x².
- Then, the equation becomes: y + 1/y = 27.
Step 2: Multiplying Through by y
To eliminate the fraction, multiply both sides by y:
- y² + 1 = 27y.
Step 3: Forming a Quadratic Equation
Reorganizing gives us a quadratic equation:
- y² - 27y + 1 = 0.
Step 4: Solving the Quadratic Equation
Using the quadratic formula, y = [27 ± √(27² - 4*1*1)] / (2*1):
- Calculate the discriminant: 729 - 4 = 725.
- Thus, y = [27 ± √725] / 2.
Step 5: Finding x² - 1
Now, we need to find x² - 1:
- Since y = x², we can express x² - 1 as (y - 1).
- Therefore, we can calculate: (y - 1) = [(27 ± √725) / 2 - 1].
Final Calculation
To simplify, we compute:
- (27 ± √725) / 2 - 1 = (25 ± √725) / 2.
This gives us the values for (x² - 1), completing our solution.
Conclusion
We find that (x² - 1) can take two possible values based on the ± sign in our calculation. Thus, the final result is represented as (25 ± √725) / 2.