Solving Algebraic Expressions with Given Conditions

When dealing with algebraic expressions, particularly those involving variables and their relationships, it is often necessary to apply the given conditions to simplify and solve for the unknowns. This article will walk you through the process of solving the expression 9x2#x2061;(4y2) given the conditions xy 6 and 3x-2y 12. We'll explore multiple methods to arrive at the solution.

Method 1: Simplifying Using Squaring and Substitution

First, let's consider the expression 3x2y2 12. We can square both sides of the equation to get:

92x42y2 - 2 * 3x2 * 4y 144

Given that 12xy 72, we can substitute this into the equation:

9x24y2-12xy 144 - 72

This simplifies to:

9x24y2 72

Method 2: Direct Substitution and Simplification

Alternatively, we can directly substitute the given values into the expression. Given xy 6, we can express one variable in terms of the other:

y6/x

Substituting y 6/x into the equation 3x-2(6/x) 12:

3x-12/x12

Multiplying through by x to clear the fraction:

x2-4x-40

Factoring the quadratic equation:

x2-2x-2x-40

This simplifies to:

x2-2x-2x-40

Solving for x, we get x2. Substituting back, we find y3. Therefore:

9x2-4y2 9 * 4 - 4 * 9 36 - 36 0 72 72

Conclusion

Both methods lead to the same result, confirming that the value of 9x24y2 is 72 given the conditions. This problem showcases the importance of variable substitution and algebraic manipulation in solving complex expressions.