Let's take a look at (x + y)^2 (x - y)^2 and (x^2 + y^2)(x^2 - y^2). While Beeker believes that these two expressions are equal for all real numbers x and y Clod believes they are not! Let's get to the bottom of this!
a) Evaluate (x + y)^2 (x - y)^2 and (x^2 + y^2)(x^2 - y^2) for x = 7 and y = 11
b) For which values of x and y does (x + y)^2 (x - y)^2 equal (x^2 + y^2)(x^2 - y^2)? For which values of x and y does (x + y)^2 (x - y)^2 not equal (x^2 + y^2)(x^2 - y^2)?
+1622 a) 18^2 * (-4)^2 and (49 + 121)(49 - 121) => 5184 and -12240
b) (x+y)(x-y)(x+y)(x-y) = (x^2 - y^2)^2.
When does (x^2 - y^2)^2 = (x^2 + y^2)(x^2 - y^2)? Divide both sides by (x^2 - y^2) => x^2 - y^2 = x^2 + y^2
Subtract x^2 from both sides, you get -y^2 = y^2, add y^2 to both sides, 2y^2 = 0, y^2 = 0, y = 0. Thus the two expressions are equal when y = 0, and x = whatever.
For which values of x and y does (x+y)62(x-y)^2 not equal (x^2 + y^2)(x^2 - y^2)? It happens when y is not 0, and x is whatever.
