The complete question is:
Find the missing x- and y-values and Pythagorean triples using the identity given:
(x² - y²)² + (2xy)² = (x² + y²)²
X Value: 4
Y Value: 3
Pythagorean Triples: ?
X Value: 5
Y Value: ?
Pythagorean Triples: (9,40,41)
X Value: ?
Y Value: 3
Pythagorean Triples: (27,36,45)
X Value: 7
Y Value: 5
Pythagorean Triples: ?
Explanation:
Pythagorean triples are three numbers p, q, r, that satisfy Pythagoras' theorem.
That is, they are numbers such that
r² = p² + q².
Given the identity
(x² - y²)² + (2xy)² = (x² + y²)²
We can say tha
p = x² - y²
q = 2xy
r = x² + y²
Now, let us use this to solve the given problems.
X Value: 4
Y Value: 3
p = 4² - 3² = 7
q = 2×3×4 = 24
r = 3² + 4² = 25
Pythagorean Triples: (7,24,25)
X Value: 5
Y Value: ? = (4)
q = 2xy = 40
2×5×y = 40
y = 40/10 = 4
Pythagorean Triples: (9,40,41)
X Value: ? = (6)
Y Value: 3
2xy = q = 36
2×3×x = 36
x = 36/6 = 6
Pythagorean Triples: (27,36,45)
X Value: 7
Y Value: 5
p = 7² - 5² = 24
q = 2×7×5 = 70
r = 7² + 5² = 74
Pythagorean Triples: ? = (24,70,74)