Algebraic equation involving unknown

Question

Here's an equation: x4+y4+z4=14. Can you solve for x, y, and z?

Answer 1

To solve the equation x^4 + y^4 + z^4 = 14, we need to find the values of x, y, and z that satisfy this equation. However, it is important to note that solving quartic equations (equations with a degree of 4) can be quite complex and often requires advanced mathematical techniques.

Unfortunately, there is no general formula for solving quartic equations like there is for quadratic equations. Instead, we typically rely on numerical methods or approximation techniques to find solutions. In this case, we will use an iterative numerical method called Newton's method to approximate the solutions.

Newton's method involves making an initial guess for the solution and then iteratively refining it until we reach a satisfactory level of accuracy. Let's outline the steps involved in applying Newton's method to solve the given equation:

Answer 2

Thinking in terms of integers and squares, you can observe that 4 + 1 + 9 = 14

so one solution to the problem given is sqrt(2), 1, and sqrt(3). These can be in any order for x, y, and z and can be negative.