To generate a Pythagorean triple using the given polynomial identity \((x^2 + y^2)^2 = (x^2 - y^2)^2 + (2xy)^2\) with \(x = 12\) and \(y = 9\), we need to calculate the corresponding values.
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Calculate \(x^2\) and \(y^2\): \[ x^2 = 12^2 = 144 \] \[ y^2 = 9^2 = 81 \]
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Calculate \(x^2 + y^2\): \[ x^2 + y^2 = 144 + 81 = 225 \]
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Calculate \(x^2 - y^2\): \[ x^2 - y^2 = 144 - 81 = 63 \]
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Calculate \(2xy\): \[ 2xy = 2 \cdot 12 \cdot 9 = 216 \]
Now we find the squares of these results:
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Left-hand side: \[ (x^2 + y^2)^2 = 225^2 = 50625 \]
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Right-hand side: \[ (x^2 - y^2)^2 = 63^2 = 3969 \] \[ (2xy)^2 = 216^2 = 46656 \]
Now, verify that the identity holds:
- Right-hand side sum: \[ 3969 + 46656 = 50625 \]
The identity holds true, and we can extract the numbers corresponding to the Pythagorean triple \( (x^2 + y^2, 2xy, x^2 - y^2) = (225, 216, 63) \).
From the provided options, the values corresponding to the generated Pythagorean triple sums include:
225
Thus, one of the values of the Pythagorean triple generated is 225.
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