Solving a Complex Equation Involving Square Roots Using Algebraic Techniques

Introduction

When dealing with complex equations that involve square roots, thorough algebraic methods can significantly simplify the process. This article explores the technique to solve the equation:

(sqrt{x} sqrt{y - 1} sqrt{z - 2} frac{1}{2}xyz)

Step-by-Step Solution

To begin, we aim to isolate the square root terms on one side of the equation:

1. Isolate the Square Root Terms:

We rewrite the equation as follows:

[sqrt{x} sqrt{y - 1} sqrt{z - 2} - frac{1}{2}xyz 0]

2. Substitutions:

To simplify the equation, we introduce the following substitutions:

- (a sqrt{x})

- (b sqrt{y - 1})

- (c sqrt{z - 2})

Using these substitutions, we can express (x), (y), and (z) in terms of (a), (b), and (c):

- (x a^2)

- (y b^2 1)

- (z c^2 2)

Substituting these into the original equation gives:

[ab sqrt{(b^2 1)(c^2 2)} frac{1}{2} (a^2)(b^2 1)(c^2 2)]

By simplifying the equation and eliminating the square root:

[ab(b^2 1)(c^2 2) frac{1}{2} (a^2)(b^2 1)(c^2 2)]

The next step is to multiply through by 2 to eliminate the fraction:

[2ab(b^2 1)(c^2 2) (a^2)(b^2 1)(c^2 2)]

This can be simplified further by distributing and dividing through by the common factors:

[2ab - a^2 - 2b^2 - 2c^2 0]

3. Completing the Square:

We complete the square for each variable:

[(a - 1)^2 (b - 1)^2 (c - 1)^2 0]

For the sum of squares to be zero, each square must individually be zero:

((a - 1)^2 0), ((b - 1)^2 0), ((c - 1)^2 0)

Therefore:

[a 1, b 1, c 1]

4. Back Substitution:

Now, we substitute back to find (x), (y), and (z):

- From (a sqrt{x} 1):

[x 1^2 1]

- From (b sqrt{y - 1} 1):

[y - 1 1 Rightarrow y 2]

- From (c sqrt{z - 2} 1):

[z - 2 1 Rightarrow z 3]

5. Solution:

The solution to the equation is:

[x 1, y 2, z 3]

Verification:

Substitute these values back into the original equation to verify that they satisfy the equation:

(sqrt{1} sqrt{2 - 1} sqrt{3 - 2} frac{1}{2} cdot 1 cdot 2 cdot 3)

(1 cdot 1 cdot 1 frac{1}{2} cdot 6)

(1 3)

The equation holds true, confirming our solution.

Conclusion

Through systematic algebraic techniques, we were able to solve a complex equation involving square roots. This method of isolating variables, substituting, simplifying, and solving step-by-step can be applied to a wide range of similar algebraic problems.