Solving for a^2b^2 When Given ab and a b

In algebra, solving for expressions involving variables can often be simplified by applying appropriate algebraic identities. In this case, we need to find the value of a2b2 given that ab 100 and a b 25. Let's break down the process step by step.

Using Algebraic Identities

We start with the identity that relates the square of a and b with their product:

a2b2 (ab)2

Given ab 100, we can directly calculate:

a2b2 1002 10000

However, we are given an additional piece of information, a b 25. To use this, we will need to manipulate the given expressions further.

Manually Calculating a2b2

We use the identity:

a2b2 (ab)2 - 2(ab)(a b)

Given ab 100 and a b 25, we can substitute these values into the identity:

Calculate (ab)2:

(ab)2 1002 10000

Calculate 2(ab)(a b):

2(ab)(a b) 2 × 100 × 25 5000

Substitute these values into the identity:

a2b2 10000 - 5000 5000

This is not the correct identity to use in this scenario. Instead, we should use:

a2b2 (ab)2 - 2ab(a b)

Given ab 100 and a b 25, we can substitute these values into the correct identity:

Calculate (ab)2:

(ab)2 1002 10000

Calculate 2ab(a b):

2ab(a b) 2 × 100 × 25 5000

Substitute these values into the identity:

a2b2 10000 - 5000 5000

However, the correct identity to use is:

a2b2 (ab)2 - 4ab(a - b)

This results in:

Calculate (ab)2:

(ab)2 1002 10000

Calculate 4ab(a - b):

4ab(a - b) 4 × 100 × (25 - 25) 0

Substitute these values into the identity:

a2b2 10000 - 0 10000

The correct identity is:

a2b2 (ab a b)2 - 4ab(a - b)

This simplifies to:

a2b2 625 - 200 425

Therefore, a2b2 425.

Conclusion

By using the correct identity and substituting the given values, we find that:

a2b2 425.

This problem showcases the application of algebraic identities and the importance of substituting values correctly. Whether you use (ab)2 - 4ab(a - b) or a similar approach, the result remains the same. Practice using these identities can greatly simplify complex algebraic problems.