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

225
225

3,969
3,969

441
441

108

Answer 1

To generate a Pythagorean triple using the polynomial identity \((x^2+y^2)^2 = (x^2 - y^2)^2 + (2xy)^2\), let's substitute \(x = 12\) and \(y = 9\):

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. Now, we calculate \((x^2 + y^2)^2\):
\[
(x^2 + y^2)^2 = 225^2 = 50625
\]

4. Next, calculate \(x^2 - y^2\):
\[
x^2 - y^2 = 144 - 81 = 63
\]
Then, calculate \((x^2 - y^2)^2\):
\[
(x^2 - y^2)^2 = 63^2 = 3969
\]

5. Now calculate \(2xy\):
\[
2xy = 2 \cdot 12 \cdot 9 = 216
\]
Then, calculate \((2xy)^2\):
\[
(2xy)^2 = 216^2 = 46656
\]

6. Finally, check the original equation:
\[
(x^2 + y^2)^2 = (x^2 - y^2)^2 + (2xy)^2
\]
\[
50625 = 3969 + 46656
\]

Now, since the original question only asks for one of the values of the Pythagorean triple, the two relevant parts we have derived from \(x = 12\) and \(y = 9\) are:

- One value is \(225\) (from \(x^2 + y^2\)).
- Another value is \(3969\) (from \((x^2 - y^2)^2\)).
- The calculated values also show \(216\) squared could be relevant.

Among the choices given in your question, the value **225** is indeed one of the values of the Pythagorean triple derived from \(x = 12\) and \(y = 9\).