Solving Equations Involving Square Roots: A Comprehensive Guide

In mathematics, solving equations that involve square roots can be a fascinating yet challenging task. One particular type of equation involves square roots in a manner that might appear as a complex system of equations but often simplifies to a straightforward solution. This article will explore such an equation and demonstrate the step-by-step process of solving it.

Understanding the Problem

The equation given is: (sqrt{x}sqrt{y-1}sqrt{z-2}frac{1}{2}xyz). This equation might initially seem challenging due to the presence of multiple variables and square roots. However, through algebraic manipulation, we can simplify and solve it.

Step-by-Step Solution

Step 1: Express Variables in Terms of Squares

Let each of the square roots be expressed as a square of a variable:

(x u^2) (sqrt{y-1}v^2) implying (y-1v^2) or (y1 v^2) (sqrt{z-2}w^2) implying (z-2w^2) or (z2 w^2)

Step 2: Substitute into the Original Equation

Substituting these into the original equation, we get:

[sqrt{u^2} cdot sqrt{1 v^2-1} cdot sqrt{2 w^2-2} frac{1}{2} u^2 (1 v^2) (2 w^2)] [u cdot v cdot w frac{1}{2} u^2 (1 v^2) (2 w^2)]

Simplify the equation:

[uvw frac{1}{2} u^2 (1 v^2) (2 w^2)]

Divide both sides by (u^2v^2w^2):

[1 frac{1}{2} u^2 (1 v^2) (2 w^2) / u^2v^2w^2] [Rightarrow 1 frac{(1 v^2) (2 w^2)}{2v^2w^2}]

Step 3: Solve for (u, v, w)

Given that:

[1 frac{(1 v^2) (2 w^2)}{2v^2w^2}]

Let's consider the simplest non-trivial solution.

The equation can be simplified to: (1 frac{1 v^2 2w^2 2v^2w^2}{2v^2w^2}) From this, we can see that the simplest solution occurs when: (v 0) and (w 0) which does not provide a valid solution. Thus, we consider (v w 1).

Substitute (v 1) and (w 1) into the original squared terms:

[v^2 - 2v 1 0 Rightarrow (v-1)^2 0 Rightarrow v 1] [w^2 - 2w 1 0 Rightarrow (w-1)^2 0 Rightarrow w 1]

Step 4: Find (u, v,) and (w)

From the simplified terms:

[v 1 Rightarrow y 1 1^2 2] [w 1 Rightarrow z 2 1^2 3] [u 1 Rightarrow x 1^2 1]

Thus, the solution is:

[x 1, y 2, z 3]

Conclusion

The equation (sqrt{x}sqrt{y-1}sqrt{z-2}frac{1}{2}xyz) is satisfied when (x1, y2, z3). This solution demonstrates the importance of algebraic manipulation and simplification in solving complex-looking equations involving square roots.