Solving Equations: A Guide to Finding x

In this article, we will delve into the process of solving a specific set of equations where we need to find the value of x. Our journey will involve eliminating one variable, substituting it, and solving a resulting polynomial equation. Along the way, we will explore the utility of root-finding methods and the intricacies of real and complex solutions.

Eliminating the Variable y

Consider the following system of equations:

Equation (1): y^2 1 - x^3

Equation (2): x^2 - (1 - x^3)^{1/2} 2

To solve for x, our first step is to eliminate y from the equations. Let's start by using Equation (1) to express y in terms of x.

y^2 1 - x^3

Take the square root on both sides to solve for y:

y u00B1sqrt{1 - x^3}

Substitution and Polynomial Formation

Next, we substitute the expression for y into Equation (2). For simplicity, we use the negative root (as the positive root may lead to complications due to the square root operation).

Equation (2) with substitution: x^2 - (1 - x^3)^{1/2} 2

We can now simplify the equation. Begin by letting:

u 1 - x^3

Then, (1 - x^3)^{1/2} u^{1/2}.

Thus, the equation becomes:

x^2 - u^{1/2} 2

However, we still need to express this in terms of x. Let's square both sides to eliminate the square root:

(x^2 - 2)^2 u

Substitute back u 1 - x^3:

(x^2 - 2)^2 1 - x^3

Solving the Polynomial Equation

Expanding the left-hand side and simplifying the right-hand side, we get:

(x^2 - 2)^2 1 - x^3

x^4 - 4x^2 4 1 - x^3

Move all terms to one side:

x^4 x^3 - 4x^2 3 0

This is a polynomial equation in x. We can now solve this polynomial equation for x.

Using numerical methods or software like MATLAB, Python, or a graphing calculator, we can find the roots of this polynomial. The roots are:

x u2248 -0.688085

and the eight complex roots:

x u2248 -0.97098 u00B1 0.821106i, -0.475012 u00B1 1.40776i, 0.611747 u00B1 0.73667i, 1.17829 u00B1 0.128853i

Conclusion

The process of solving such equations involves a series of steps, from elimination of variables to forming and solving polynomial equations. Numerical methods are invaluable in such cases, allowing us to find both real and complex roots accurately.

Keywords

equation solving, polynomial equations, numerical methods