If \(x - y = 7\) and \(xy = 60\), then what is the value of \(x^2 + y^2\)?
169
We are given \(x - y = 7\) and \(xy = 60\), and we need \(x^2 + y^2\).
The key identity connects a difference with a sum of squares: \((x - y)^2 = x^2 - 2xy + y^2\).
Rearranging this gives \(x^2 + y^2 = (x - y)^2 + 2xy\), so we only need the two quantities already supplied.
Substitute the values: \(x^2 + y^2 = (7)^2 + 2(60)\).
Evaluate each term: \((7)^2 = 49\) and \(2 \times 60 = 120\).
Add them: \(49 + 120 = 169\).
Using the square-of-a-difference identity avoids solving for \(x\) and \(y\) separately. The value of \(x^2 + y^2\) is 169.
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