If a = b and b > 0 and a > 0, then 1 = 2 a>0 and b>0 a = b ab = b2 ab - a2 = b2 - a2 a(b-a) = (b+a)(b-a) a = (b+a) a = a + a (substituting a for b) a = 2a a/a = 2a/a 1 = 2 a = x [true for some a's and x's] a+a = a+x [add a to both sides] 2a = a+x [a+a = 2a] 2a-2x = a+x-2x [subtract 2x from both sides] 2(a-x) = a+x-2x [2a-2x = 2(a-x)] 2(a-x) = a-x [x-2x = -x] 2 = 1 [divide both sides by a-x] a = b a2 = ab a2 - b2 = ab - b2 (a + b)(a - b) = b(a - b) a + b = b 2b = b 2 = 1 (x + 1)2 = x2 + 2x + 1 (x + 1)2 - (2x + 1) = x2 (x + 1)2 - (2x + 1) - x(2x + 1) = x2 - x(2x + 1) (x + 1)2 - (x + 1)(2x + 1) + (1/4)(2x + 1)2 = x2 - x(2x + 1) + (1/4)(2x + 1)2 [(x + 1) - (1/2)(2x + 1)]2 = [x - (1/2)(2x + 1)]2 (x + 1) - (1/2)(2x + 1) = x - (1/2)(2x + 1) x + 1 = x 1 = 0
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