Theorem : Suppose x and y are distinct integers. Then (x+1)^2=(y+1)^2. If and only if x + y = - 2

Feb 3rd, 2012
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Theorem : Suppose x and y are distinct integers. Then (x+1)^2=(y+1)^2. If and only if x + y = - 2

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Theorem : Suppose x and y are distinct integers. Then (x+1)^2=(y+1)^2.If and only if x + y = - 2Proof : Consider (x+1)2=(y+1)2We expand it, thenx2 + 2x + 1 = y2 + 2y +1Nowx2 + 2x = y2 + 2yx2 - y2 = 2y - 2xx2 - y2 = - 2x + 2yx2 - y2 = - 2(x - y)Now(x - y)(x + y) = - 2(x - y)Now we divide by (x - y), we get(x - y)(x + y)/(x - y) = - 2(x - y)/(x - y)(x + y) = - 2Hence we proved that x and y are two distinct integers if (x + 1)2 = (y + 1)2Now we prove converselyConsider x + y = -2Now we multilpy by (x - y) on both sides, we get(x - y)(x + y) = - 2(x - y)x2 - y2 = - 2x +

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