Math Problem Statement

Given sqrt(x^2-4)-1+(x-sqrt(x^2-4))^k (x+sqrt(x^2-4))^-k=0, please express x in terms of k,. You may use hyperbolic function

Solution

To express xx in terms of kk for the equation

x241+(xx24)k(x+x24)k=0,\sqrt{x^2 - 4} - 1 + \left(x - \sqrt{x^2 - 4}\right)^k \left(x + \sqrt{x^2 - 4}\right)^{-k} = 0,

we can utilize hyperbolic functions to simplify the terms involving square roots.

Step 1: Express xx in terms of hyperbolic functions

Notice that x24x^2 - 4 inside the square root can be related to hyperbolic identities. Let’s define: x=2cosh(θ).x = 2 \cosh(\theta). Then: x24=4cosh2(θ)4=4(cosh2(θ)1)=4sinh2(θ),x^2 - 4 = 4 \cosh^2(\theta) - 4 = 4 (\cosh^2(\theta) - 1) = 4 \sinh^2(\theta), so that x24=2sinh(θ).\sqrt{x^2 - 4} = 2 \sinh(\theta).

Substituting these expressions, we have: x24=2sinh(θ)andx=2cosh(θ).\sqrt{x^2 - 4} = 2 \sinh(\theta) \quad \text{and} \quad x = 2 \cosh(\theta).

Step 2: Simplify the expression

Substitute into the equation: 2sinh(θ)1+(2cosh(θ)2sinh(θ))k(2cosh(θ)+2sinh(θ))k=0.2 \sinh(\theta) - 1 + \left(2 \cosh(\theta) - 2 \sinh(\theta)\right)^k \left(2 \cosh(\theta) + 2 \sinh(\theta)\right)^{-k} = 0.

Factor out constants from the terms in parentheses: 2sinh(θ)1+(2(cosh(θ)sinh(θ)))k(2(cosh(θ)+sinh(θ)))k=0.2 \sinh(\theta) - 1 + \left(2 (\cosh(\theta) - \sinh(\theta))\right)^k \left(2 (\cosh(\theta) + \sinh(\theta))\right)^{-k} = 0.

Simplify further: 2sinh(θ)1+2k(cosh(θ)sinh(θ)cosh(θ)+sinh(θ))k=0.2 \sinh(\theta) - 1 + 2^k \left(\frac{\cosh(\theta) - \sinh(\theta)}{\cosh(\theta) + \sinh(\theta)}\right)^k = 0.

Using hyperbolic identities, cosh(θ)±sinh(θ)=e±θ\cosh(\theta) \pm \sinh(\theta) = e^{\pm \theta}, we get: 2sinh(θ)1+2k(eθeθ)k=0,2 \sinh(\theta) - 1 + 2^k \left(\frac{e^{-\theta}}{e^{\theta}}\right)^k = 0, which simplifies to: 2sinh(θ)1+2ke2kθ=0.2 \sinh(\theta) - 1 + 2^k e^{-2k\theta} = 0.

Step 3: Solve for θ\theta

Rearrange to isolate e2kθe^{-2k\theta}: 2sinh(θ)1=2ke2kθ.2 \sinh(\theta) - 1 = -2^k e^{-2k\theta}. From here, you can solve for θ\theta in terms of kk, and then find x=2cosh(θ)x = 2 \cosh(\theta).

Would you like further simplification or assistance with specific steps?


Here are five related questions for exploration:

  1. How can hyperbolic functions simplify other types of expressions involving square roots?
  2. What is the relationship between hyperbolic and exponential functions in general problem-solving?
  3. How would the solution change if the exponent kk had different values?
  4. Are there alternative trigonometric substitutions that could work similarly to hyperbolic substitutions in this context?
  5. How does this method compare to solving such an equation without using hyperbolic functions?

Tip: Hyperbolic identities can be a powerful tool for simplifying expressions that involve square roots, particularly when terms like x2a2x^2 - a^2 appear.

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Math Problem Analysis

Mathematical Concepts

Hyperbolic Functions
Algebraic Manipulation
Exponential Functions

Formulas

x = 2 cosh(θ)
sqrt(x^2 - 4) = 2 sinh(θ)
(cosh(θ) ± sinh(θ) = e^±θ)

Theorems

Hyperbolic Identity
Exponential Property for Hyperbolic Functions

Suitable Grade Level

Grades 11-12