Solving the Equation ( frac{ab}{sqrt{ab}} frac{2}{1} ): Techniques and Insights

The equation ( frac{ab}{sqrt{ab}} frac{2}{1} ) may seem complex, but through careful algebraic manipulation, we can simplify and solve it step-by-step. This article will guide you through the process and provide insights into similar algebraic problems.

Initial Setup and Simplification

Let's start with the given equation:

( frac{ab}{sqrt{ab}} frac{2}{1} )

By applying cross multiplication, we get:

( ab 2sqrt{ab} )

Eliminating the Square Root

To remove the square root, we can square both sides of the equation:

( (ab)^2 (2sqrt{ab})^2 )

Expanding both sides gives us:

( a^2b^2 4ab )

Next, we rearrange the equation to isolate terms involving (a) and (b):

( a^2b^2 - 4ab 0 )

Factorization and Simplification

Factoring out (ab) from the terms on the left-hand side, we get:

( ab(ab - 4) 0 )

This factorization gives us two potential solutions:

( ab 0 ) or ( ab - 4 0 )

Since (a) and (b) are typically not zero in this type of equation, we focus on:

( ab - 4 0 )

Solving for (ab), we get:

( ab 4 )

Introducing Variables for Simplification

Introduce the variable (x frac{a}{b}). This allows us to express (a) in terms of (b):

( a xb )

Substituting (a xb) into our original equation:

( frac{xb cdot b}{sqrt{xb cdot b}} 2 )

This simplifies to:

( xb cdot b 2sqrt{xb cdot b} )

Further simplification gives:

( x cdot b^2 2b sqrt{x} )

Dividing both sides by (b) (assuming (b eq 0)), we get:

( x cdot b 2sqrt{x} )

Now, squaring both sides to eliminate the square root:

( x^2 cdot b^2 4x )

Dividing both sides by (b^2) (assuming (b eq 0)), we get:

( x^2 - 4x 0 )

Factoring out (x), we get:

( x(x - 4) 0 )

This gives us two solutions:

( x 0 ) or ( x 4 )

Since (x frac{a}{b}) and (x 0) is not a valid solution in this context, we have:

( x 1 )

Therefore, we conclude that:

( frac{a}{b} 1 )

Conclusion

The equation ( frac{ab}{sqrt{ab}} frac{2}{1} ) simplifies to ( frac{a}{b} 1 ) when solved through algebraic manipulation and substitution. This result means that (a) and (b) are equal, and their ratio ( frac{a}{b} ) is 1.

The process of solving this equation demonstrates the power of algebraic manipulation, including introducing variables, squaring both sides, and factoring, in solving complex equations involving square roots.

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