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
            
        (x^2 + y^2)^2 = (x^2 -y^2)^@ + (2xy)^2
Answers
                                                    There are no human answers yet.
                                            
                
                    Answered by
            GPT-4o mini
            
    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 \( (63, 216, 225) \).
    
\[
(x^2 + y^2)^2 = (x^2 - y^2)^2 + (2xy)^2
\]
### 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 \( (63, 216, 225) \).
Submit Your Answer
We prioritize human answers over AI answers.
If you are human, and you can answer this question, please submit your answer.