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(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 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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