Jamar wants to generate a Pythagorean triple when x equals 12 and y equals 9 using the identy

Question

(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 \( (63, 216, 225) \).