Janel wants to generate a Pythagorean triple when x equals 18 and y equals 9 using the polynomial identity (x2+y2)2=(x2−y2)2+(2xy)2
. Which of the following is one of the values of the Pythagorean triple?(1 point)
Responses
162
162
324
324
81
81
729
Using the polynomial identity, we have:
(x^2 + y^2)^2 = (x^2 - y^2)^2 + (2xy)^2
(18^2 + 9^2)^2 = (18^2 - 9^2)^2 + (2*18*9)^2
(405)^2 = (243)^2 + (324)^2
165025 = 59049 + 104976
165025 = 163025
So, the correct answer is 324.
To generate a Pythagorean triple when x equals 18 and y equals 9 using the polynomial identity (x^2+y^2)^2=(x^2−y^2)^2+(2xy)^2, we can substitute the given values into the equation:
(18^2 + 9^2)^2 = (18^2 - 9^2)^2 + (2 * 18 * 9)^2
Simplifying further:
(324 + 81)^2 = (324 - 81)^2 + (36)^2
(405)^2 = (243)^2 + 1296
163,025 = 59,049 + 1,296
163,025 = 60,345
Therefore, none of the given options is a value of the Pythagorean triple.
To find the values of the Pythagorean triple using the given polynomial identity, we can substitute the values of x and y into the equation and solve for the resulting expression.
Given: x = 18, y = 9
Substituting these values into the polynomial identity:
(x^2 + y^2)^2 = (x^2 - y^2)^2 + (2xy)^2
(18^2 + 9^2)^2 = (18^2 - 9^2)^2 + (2 * 18 * 9)^2
Simplifying:
(324 + 81)^2 = (324 - 81)^2 + (36)^2
(405)^2 = (243)^2 + (36)^2
Now we can compare the left side of the equation to see if any of the given options match.
Option 1: 162
Option 2: 324
Option 3: 81
Option 4: 729
Checking the options:
(162)^2 ≠ (243)^2 + (36)^2
(324)^2 = (243)^2 + (36)^2
(81)^2 ≠ (243)^2 + (36)^2
(729)^2 ≠ (243)^2 + (36)^2
So, the correct answer is 324, which matches the right side of the equation.