Squaring May Help

Algebra Level 3

If x − 1 − x + 1 + 1 = 0 \sqrt{x-1}-\sqrt{x+1}+1=0 , then what is the value of 4 x 4x ?


The answer is 5.

Satwik Murarka by Satwik Murarka , 20, India

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2 solutions

Viki Zeta
Sep 26, 2016

x − 1 − x + 1 + 1 = 0 x − 1 + 1 = x + 1 ( x − 1 + 1 ) 2 = ( x + 1 ) 2 x − 1 + 1 + 2 x − 1 = x + 1 1 = 2 x − 1 1 = 2 x − 1 1 = 4 ( x − 1 ) 1 = 4 x − 4 4 x = 5 \sqrt{x-1}-\sqrt{x+1}+1=0 \\ \sqrt{x-1}+1=\sqrt{x+1} \\ (\sqrt{x-1}+1)^2 = (\sqrt{x+1})^2 \\ x - 1 + 1 + 2\sqrt{x-1} = x + 1 \\ 1 = 2\sqrt{x-1} \\ 1 = 2\sqrt{x-1} \\ 1 = 4(x-1) \\ 1 = 4x - 4 \\ 4x = 5

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Nice solution.Did it the same way.

Satwik Murarka - 4 years, 8 months ago
Chew-Seong Cheong
Sep 27, 2016

x − 1 − x + 1 + 1 = 0 Rearranging x + 1 − x − 1 = 1 Squaring both sides x + 1 − 2 x 2 − 1 + x − 1 = 1 2 x − 1 = 2 x 2 − 1 Squaring both sides 4 x 2 − 4 x + 1 = 4 x 2 − 4 ⟹ 4 x = 5 \begin{aligned} \sqrt{x-1}-\sqrt{x+1} + 1 & = 0 & \small \color{#3D99F6}{\text{Rearranging}} \\ \sqrt{x+1}-\sqrt{x-1} & = 1 & \small \color{#3D99F6}{\text{Squaring both sides}} \\ x+1 - 2\sqrt{x^2-1} + x-1 & = 1 \\ 2x - 1& = 2\sqrt{x^2-1} & \small \color{#3D99F6}{\text{Squaring both sides}} \\ 4x^2 - 4x +1 & = 4x^2 - 4 \\ \implies 4x & = \boxed{5} \end{aligned}

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