If \(x + y = 7\) and \(xy = 12\), calculate the value of \(x^{2} + y^{2}\).
25
Given: \(x + y = 7\) and \(xy = 12\).
Use the algebraic identity \((x+y)^{2}=x^{2}+y^{2}+2xy\).
Rearrange to get \(x^{2}+y^{2}=(x+y)^{2}-2xy\).
Substitute the given values: \(x^{2}+y^{2}=(7)^{2}-2(12)\).
\(=49-24\).
\(=25\).
Hence the value of \(x^{2}+y^{2}\) is 25.
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