Question

find the missing x- and y-values and pythagorean triples using the identity ((x^2 - y^2)^2 + (2xy)^2 = (x^2 + y^2)^2).
write the triples in parentheses with commas but no spaces between the values, and order the valu from least to greatest.
type the correct answer in each box.
(the table has columns: x - value, y - value, pythagorean triple. rows: 4, 3, blank; 5, blank, (9, 40, 41); blank, 3, (27, 36, 45); 7, 5, blank)

Explanation:

Step1: Calculate triple for x=4,y=3

Compute $x^2-y^2=4^2-3^2=16-9=7$, $2xy=2*4*3=24$, $x^2+y^2=4^2+3^2=16+9=25$. Order: 7,24,25.

Step2: Find y for x=5, triple (9,40,41)

Use $2xy=40$: $2*5*y=40 \implies 10y=40 \implies y=4$. Verify $x^2-y^2=25-16=9$, $x^2+y^2=25+16=41$.

Step3: Find x for y=3, triple (27,36,45)

Use $2xy=36$: $2*x*3=36 \implies 6x=36 \implies x=6$. Verify $x^2-y^2=36-9=27$, $x^2+y^2=36+9=45$.

Step4: Calculate triple for x=7,y=5

Compute $x^2-y^2=49-25=24$, $2xy=2*7*5=70$, $x^2+y^2=49+25=74$. Order:24,70,74.

Answer:

x-value	y-value	Pythagorean Triple
5	4	(9,40,41)
6	3	(27,36,45)
7	5	(24,70,74)