Question

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

Answer 1

To generate a Pythagorean triple using the given identity, we need to use the values \( x = 12 \) and \( y = 9 \) in the formula:

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

### Step 1: Calculate \( x^2 \) and \( y^2 \)

1. Calculate \( x^2 \):
\[
x^2 = 12^2 = 144
\]

2. Calculate \( y^2 \):
\[
y^2 = 9^2 = 81
\]

### Step 2: Calculate \( x^2 + y^2 \) and \( x^2 - y^2 \)

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

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

### Step 3: Calculate \( 2xy \)

1. Calculate \( 2xy \):
\[
2xy = 2 \times 12 \times 9 = 216
\]

### Step 4: Substitute into the identity

1. We now substitute these values into the formula:
\[
(x^2 + y^2)^2 = 225^2
\]
\[
(x^2 - y^2)^2 + (2xy)^2 = 63^2 + 216^2
\]

### Step 5: Calculate the squares

1. Calculate \( 225^2 \):
\[
225^2 = 50625
\]

2. Calculate \( 63^2 \):
\[
63^2 = 3969
\]

3. Calculate \( 216^2 \):
\[
216^2 = 46656
\]

### Step 6: Compare both sides

Now, calculate the right side:
\[
63^2 + 216^2 = 3969 + 46656 = 50625
\]

### Conclusion

Since both sides are equal:
\[
(x^2 + y^2)^2 = (x^2 - y^2)^2 + (2xy)^2
\]
\[
50625 = 50625
\]

This confirms the identity holds.

### Deriving the Pythagorean triple

Using the values from above:
- The Pythagorean triple can be derived as:
- \( a = x^2 - y^2 = 63 \)
- \( b = 2xy = 216 \)
- \( c = x^2 + y^2 = 225 \)

Thus, the Pythagorean triple is \( (63, 216, 225) \).