Question

Jamar wants to generate a Pythagorean triple when x equals 12 and y equals 9 using the identity
(x^2 + y^2)^2 = (x^2 -y^2)^@ + (2xy)^2

answer options
108
225
3969
441
which is correct

Answer 1

To generate a Pythagorean triple using the identity \((x^2 + y^2)^2 = (x^2 - y^2)^2 + (2xy)^2\), we need to substitute \(x = 12\) and \(y = 9\) into the formula.

First, we calculate \(x^2\) and \(y^2\):

\[
x^2 = 12^2 = 144
\]
\[
y^2 = 9^2 = 81
\]

Now, we can compute \(x^2 + y^2\) and \(x^2 - y^2\):

\[
x^2 + y^2 = 144 + 81 = 225
\]
\[
x^2 - y^2 = 144 - 81 = 63
\]

Next, we calculate \(2xy\):

\[
2xy = 2 \times 12 \times 9 = 216
\]

Now we will square the values:

1. \((x^2 + y^2)^2\):

\[
(225)^2 = 50625
\]

2. \((x^2 - y^2)^2\):

\[
(63)^2 = 3969
\]

3. \((2xy)^2\):

\[
(216)^2 = 46656
\]

Now, we check the identity:

\[
(x^2 + y^2)^2 = (x^2 - y^2)^2 + (2xy)^2
\]

Plugging in the values we calculated:

\[
50625 = 3969 + 46656
\]

Calculating the right-hand side:

\[
3969 + 46656 = 50625
\]

The identity holds, and now we look at the options provided:

- 108
- 225
- 3969
- 441

Among the options, \(3969\) is the result we calculated for \((x^2 - y^2)^2\). Thus, the correct answer from the options is:

**3969**