Question

Jamar wants to generate a Pythagorean triple when x equals 12 and y equals 9 using the polynomial identity (x2+y2)2=(x2−y2)2+(2xy)2
. Which of the following is one of the values of the Pythagorean triple?(1 point)
Responses

441
441

3,969
3,969

108
108

225

Answer 1

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.

1. Calculate \(x^2\) and \(y^2\):
\[
x^2 = 12^2 = 144
\]
\[
y^2 = 9^2 = 81
\]

2. Calculate \(x^2 + y^2\):
\[
x^2 + y^2 = 144 + 81 = 225
\]

3. Calculate \(x^2 - y^2\):
\[
x^2 - y^2 = 144 - 81 = 63
\]

4. Calculate \(2xy\):
\[
2xy = 2 \cdot 12 \cdot 9 = 216
\]

Now we find the squares of these results:

- Left-hand side:
\[
(x^2 + y^2)^2 = 225^2 = 50625
\]

- 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**.