Given Equations
We start with the equations:
- a - b = 7
- ab = 9
Objective
We need to find the value of a²b².
Using the Relationships
To find a²b², we can use the relationship:
- a²b² = (ab)²
This means we need to square the product of a and b.
Calculating (ab)²
Since we know that ab = 9:
- (ab)² = 9²
- (ab)² = 81
Thus, a²b² = 81.
Verification of Values
To ensure that the values of a and b satisfy the original equations, we can find a and b:
1. From a - b = 7, we can express a as:
- a = b + 7
2. Substitute a into the second equation:
- (b + 7)b = 9
- b² + 7b - 9 = 0
Using the Quadratic Formula
Now, we can solve for b using the quadratic formula:
- b = [-7 ± √(7² - 4(1)(-9))] / 2(1)
- b = [-7 ± √(49 + 36)] / 2
- b = [-7 ± √85] / 2
This gives us two possible values for b.
Finding a
Once we have b, we can find a using:
- a = b + 7
Now we have both a and b, and we can confirm that:
- a²b² = 81 is indeed valid.
Conclusion
Thus, the value of a²b² is:
- 81.