Infinite solutions?

Algebra Level 3

x + 8 + 6 x − 1 − x + 3 + 4 x − 1 = 1 \sqrt{ x+8+6\sqrt{x-1} }-\sqrt{ x+3+4\sqrt{x-1}}=1

If the solution set for the above equation for x ∈ R x\in \mathbb{R} is in the for [ a , ∞ ) [a,\infty) , find a a .


The answer is 1.

Akshat Sharda by Akshat Sharda , 21, India

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1 solution

Sabhrant Sachan
Jun 30, 2016

x + 8 + 6 x − 1 = ( x − 1 ) + ( 9 ) + 2 ⋅ 3 ⋅ x − 1 = ( x − 1 + 3 ) 2 x + 3 + 4 x − 1 = ( x − 1 ) + ( 4 ) + 2 ⋅ 2 ⋅ x − 1 = ( x − 1 + 2 ) 2 ⟹ x − 1 + 3 − x − 1 − 2 = 1 x ∈ [ 1 , ∞ ) → Domain of our original equation \quad x+8+6\sqrt{x-1} = (x-1)+(9)+2\cdot3\cdot\sqrt{x-1} = \left( \sqrt{x-1}+3 \right)^2 \\ \quad x+3+4\sqrt{x-1} = (x-1)+(4)+2\cdot2\cdot\sqrt{x-1} = \left( \sqrt{x-1}+2 \right)^2 \\ \quad \implies \sqrt{x-1}+3-\sqrt{x-1}-2 = \boxed{1} \\ \quad \boxed{x \in \left[ 1 , \infty \right)} \quad \quad \color{#20A900}{ \rightarrow \text{ Domain of our original equation}}

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Good question!

Deepak Kumar - 4 years, 11 months ago

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