Let's focus on finding the Pythagorean triples for the given [tex]\( x \)[/tex]- and [tex]\( y \)[/tex]-values using the identity:
[tex]\[
(x^2 - y^2)^2 + (2xy)^2 = (x^2 + y^2)^2
\][/tex]
We can calculate the Pythagorean triple [tex]\((a, b, c)\)[/tex] as follows:
- [tex]\(a = x^2 - y^2\)[/tex]
- [tex]\(b = 2xy\)[/tex]
- [tex]\(c = x^2 + y^2\)[/tex]
We then sort [tex]\(a\)[/tex], [tex]\(b\)[/tex], and [tex]\(c\)[/tex] in ascending order. Let’s perform these calculations for the specified pairs.
### For [tex]\( x = 4 \)[/tex] and [tex]\( y = 3 \)[/tex]:
1. Calculate [tex]\( a \)[/tex]:
[tex]\[
a = x^2 - y^2 = 4^2 - 3^2 = 16 - 9 = 7
\][/tex]
2. Calculate [tex]\( b \)[/tex]:
[tex]\[
b = 2xy = 2 \cdot 4 \cdot 3 = 24
\][/tex]
3. Calculate [tex]\( c \)[/tex]:
[tex]\[
c = x^2 + y^2 = 4^2 + 3^2 = 16 + 9 = 25
\][/tex]
4. Order [tex]\(a\)[/tex], [tex]\(b\)[/tex], and [tex]\(c\)[/tex]:
[tex]\[
(7, 24, 25)
\][/tex]
So the Pythagorean triple for [tex]\( x = 4 \)[/tex] and [tex]\( y = 3 \)[/tex] is [tex]\((7, 24, 25)\)[/tex].
### For [tex]\( x = 7 \)[/tex] and [tex]\( y = 3 \)[/tex]:
1. Calculate [tex]\( a \)[/tex]:
[tex]\[
a = x^2 - y^2 = 7^2 - 3^2 = 49 - 9 = 40
\][/tex]
2. Calculate [tex]\( b \)[/tex]:
[tex]\[
b = 2xy = 2 \cdot 7 \cdot 3 = 42
\][/tex]
3. Calculate [tex]\( c \)[/tex]:
[tex]\[
c = x^2 + y^2 = 7^2 + 3^2 = 49 + 9 = 58
\][/tex]
4. Order [tex]\(a\)[/tex], [tex]\(b\)[/tex], and [tex]\(c\)[/tex]:
[tex]\[
(40, 42, 58)
\][/tex]
So the Pythagorean triple for [tex]\( x = 7 \)[/tex] and [tex]\( y = 3 \)[/tex] is [tex]\((40, 42, 58)\)[/tex].
Thus, the completed table will be:
[tex]\[
\begin{tabular}{|c|c|c|}
\hline
$x$-value & $y$-value & Pythagorean Triple \\
\hline
4 & 3 & (7,24,25) \\
\hline
5 & 3 & (9,40,41) \\
\hline
7 & 5 & (27,36,45) \\
\hline
7 & 3 & (40,42,58) \\
\hline
\end{tabular}
\][/tex]
