Solving for ( x^2 - y^2 ) Given ( xy 7 ) and ( xy 12 )

The given problem involves finding the value of ( x^2 - y^2 ) given that ( xy 7 ) and ( xy 12 ). Initially, the problem seems contradictory, as it presents two identical expressions for ( xy ), but to solve it, we will proceed with the given steps, which involve the use of identities and algebraic manipulation.

Step 1: Use the identity for the difference of squares:

Recall the identity: ( x^2 - y^2 (x y)(x - y) ). This is the first key point.

Step 2: Express ( x^2 - y^2 ) using the given information.

From the problem, we know ( xy 7 ) and ( xy 12 ), indicating an inconsistency, but let's assume the correct interpretation is that there are two distinct equations or values of ( xy ).

Solving the Quadratic Equation

First, we solve the quadratic equation derived from the given conditions:

( x^2 - y^2 7 ) and ( xy 12 ).

This can be achieved by forming a quadratic equation based on the given conditions:

( x^2 - 7x 12 0 ).

Solving the Quadratic Equation

We solve the equation ( x^2 - 7x 12 0 ) using the quadratic formula:

( x frac{-b pm sqrt{b^2 - 4ac}}{2a} )

Here, ( a 1 ), ( b -7 ), and ( c 12 ).

( x frac{7 pm sqrt{49 - 48}}{2} )

( x frac{7 pm 1}{2} )

This gives us two solutions:

( x 4 ) or ( x 3 ).

Verifying Values

Using ( x 3 ) and ( x 4 ), we find ( y ) values by substituting back into ( xy 12 ).

( y frac{12}{3} 4 ) or ( y frac{12}{4} 3 ).

Using the Identity

Now, we know the pairs ((x, y)) are ((3, 4)) and ((4, 3)).

Using the identity ( x^2 - y^2 (x y)(x - y) ), we can calculate ( x^2 - y^2 ) for both pairs:

Pairs and Calculations:

For ( (x, y) (3, 4) ):

( x^2 - y^2 (3 4)(3 - 4) 7 cdot (-1) -7 )

For ( (x, y) (4, 3) ):

( x^2 - y^2 (4 3)(4 - 3) 7 cdot 1 7 )

Conclusion

Thus, the value of ( x^2 - y^2 ) is either ( 7 ) or ( -7 ).

In a practical sense, both solutions exist due to the nature of the algebraic manipulation and the properties of the difference of squares.

Final Answer: ( x^2 - y^2 pm 7 )