Question

Medal and fan! :)
A student is asked to calculate the Pythagorean Triples for the number 11 using the identity (x2 − y2)2 + (2xy)2 = (x2 + y2)2. The student's steps are shown below.

Step 1: 11 = 62 − 52; therefore, x = 6 and y = 5

Step 2: (62 − 52)2 + (2 ⋅ 6 ⋅ 5)2 = (62 + 52)2

Step 3: (a − b)2 + (60)2 = (36 + 25)2

Step 4: (11)2 + (60)2 = (c)2

Step 5: The Pythagorean Triples are 11, 60, and 61.

In Steps 3 and 4, what are the values of a, b, and c, respectively?

Answer 1

@Photon336

Answer 2

Think you can help? :) please

Answer 3

Should this be \( (a − b)^2 + (60)^2 = (36 + 25)^2\)?

Answer 4

We can prove the identity (x2 + y2)2 = (x2 – y2)2 + (2xy)2 by manipulating the equation step by step, showing our work as we go. We'll change only one side of the equation and see if we can change it into the other side.

(x2 + y2)2
= (x2 + y2)(x2 + y2)
= x4 + x2y2 + y2x2 + y4
= x4 + 2x2y2 + y4
= x4 – 2x2y2 + 4x2y2 + y4
= (x4 – 2x2y2 + y4) + 4x2y2
= (x2 – y2)2 + (2xy)2

There. Since both sides are equal, we can call this equation an identity. The identity is true for all values of x and y. Now, i dont understand what the steps in question means..

Answer 5

Compare step 2 and 3.

How did step 2 become step 3? Focus on the LHS in particular, comparing what a and b are with step 2.

Step 4 comes from step 3. So what part of step 3 is equivalent to c^2?