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Mathematics

Find the square of:

2x+1x+12x +\dfrac{1}{x}+ 1

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Answer

(2x+1x+1)(2x +\dfrac{1}{x}+ 1)2

Using the formula

(∵ (x + y + z)2 = x2 + y2 + z2 + 2xy + 2yz + 2xz)

=(2x)2+(1x)2+12+2×2x×(1x)+2×(1x)×1+2×1×2x=4x2+(1x2)+1+(4xx)+(2x)+4x=4x2+(1x2)+1+4+(2x)+4x=4x2+(1x2)+5+(2x)+4x= (2x)^2 + \Big(\dfrac{1}{x}\Big)^2 + 1^2 + 2 \times 2x \times \Big(\dfrac{1}{x}\Big) + 2 \times \Big(\dfrac{1}{x}\Big) \times 1 + 2 \times 1 \times 2x\\[1em] = 4x^2 + \Big(\dfrac{1}{x^2}\Big) + 1 + \Big(\dfrac{4x}{x}\Big) + \Big(\dfrac{2}{x}\Big) + 4x\\[1em] = 4x^2 + \Big(\dfrac{1}{x^2}\Big) + 1 + 4 + \Big(\dfrac{2}{x}\Big) + 4x\\[1em] = 4x^2 + \Big(\dfrac{1}{x^2}\Big) + 5 + \Big(\dfrac{2}{x}\Big) + 4x

Hence, (2x+1x+1)(2x +\dfrac{1}{x}+ 1)2 = 4x2 + (1x2)\Big(\dfrac{1}{x^2}\Big) + 5 + (2x)\Big(\dfrac{2}{x}\Big) + 4x

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