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If a = b and b > 0 and a > 0, then 1 = 2 a>0 and b>0 a = b ab = b^{2} ab  a^{2} = b^{2}  a^{2} a(ba) = (b+a)(ba) 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] 2a2x = a+x2x [subtract 2x from both sides] 2(ax) = a+x2x [2a2x = 2(ax)] 2(ax) = ax [x2x = x] 2 = 1 [divide both sides by ax] a = b a^{2} = ab a^{2}  b^{2} = ab  b^{2} (a + b)(a  b) = b(a  b) a + b = b 2b = b 2 = 1 (x + 1)^{2} = x^{2} + 2x + 1 (x + 1)^{2}  (2x + 1) = x^{2} (x + 1)^{2}  (2x + 1)  x(2x + 1) = x^{2}  x(2x + 1) (x + 1)^{2}  (x + 1)(2x + 1) + (1/4)(2x + 1)^{2} = x^{2}  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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