Question

using the polynomiak identity (x^2+y^2)^2=(x^2-y^2+(2xy)^2 to generate a pythagereorm triple where x equals 10 and y equals 6 which of the following is one of the values of the pythagereom theorem
A.120
B.16
C.256
D.60

Answer 1

Are you sure

Answer 2

To generate a Pythagorean triple using the polynomial identity \((x^2 + y^2)^2 = (x^2 - y^2)^2 + (2xy)^2\), we will substitute \(x = 10\) and \(y = 6\).

1. First, calculate \(x^2\) and \(y^2\):
\[
x^2 = 10^2 = 100
\]
\[
y^2 = 6^2 = 36
\]

2. Now, calculate \(x^2 + y^2\):
\[
x^2 + y^2 = 100 + 36 = 136
\]

3. Next, calculate \(x^2 - y^2\):
\[
x^2 - y^2 = 100 - 36 = 64
\]

4. Then calculate \(2xy\):
\[
2xy = 2 \cdot 10 \cdot 6 = 120
\]

Now we have:
- \( (x^2 - y^2) = 64 \)
- \( (2xy) = 120 \)

According to the identity:
\[
(x^2 + y^2)^2 = (x^2 - y^2)^2 + (2xy)^2
\]
We can confirm that:
\[
136^2 = 64^2 + 120^2
\]

Calculating both sides:
- Left side:
\[
136^2 = 18496
\]

- Right side:
\[
64^2 = 4096 \quad \text{and} \quad 120^2 = 14400
\]
\[
64^2 + 120^2 = 4096 + 14400 = 18496
\]

Both sides match, confirming that the identity holds.

The values of the Pythagorean triple generated are:
\[
a = x^2 - y^2 = 64, \quad b = 2xy = 120, \quad c = x^2 + y^2 = 136
\]

Now, checking the options:
- A. 120 (one of the values)
- B. 16
- C. 256
- D. 60

Thus, the answer is **A. 120**.

Answer 3

Yes, let's go through it again step by step to ensure the calculations are correct and clarify the result:

We use the polynomial identity:
\[
(x^2 + y^2)^2 = (x^2 - y^2)^2 + (2xy)^2
\]
with \(x = 10\) and \(y = 6\).

1. Calculate \(x^2\) and \(y^2\):
\[
x^2 = 10^2 = 100
\]
\[
y^2 = 6^2 = 36
\]

2. Calculate \(x^2 + y^2\):
\[
x^2 + y^2 = 100 + 36 = 136
\]

3. Calculate \(x^2 - y^2\):
\[
x^2 - y^2 = 100 - 36 = 64
\]

4. Calculate \(2xy\):
\[
2xy = 2 \cdot 10 \cdot 6 = 120
\]

So, we have:
- \(x^2 + y^2 = 136\)
- \(x^2 - y^2 = 64\)
- \(2xy = 120\)

Now, we can confirm the Pythagorean triple:
- The expression states that \( (x^2 + y^2)^2 = (x^2 - y^2)^2 + (2xy)^2 \).
- We already calculated:
- Left side:
\[
(136)^2 = 18496
\]
- Right side:
\[
(64)^2 + (120)^2 = 4096 + 14400 = 18496
\]

So, both sides equal \(18496\), confirming this identity is valid.

The values of the Pythagorean triple are:
- \(a = 64\)
- \(b = 120\)
- \(c = 136\)

Looking at your original options:
- A. 120 (this is one of the values)
- B. 16
- C. 256
- D. 60

Thus, indeed, **the answer is A. 120**. It represents one of the numeric components of the generated Pythagorean triple. If there is any specific part you want clarification on, please let me know!