Simple Form

Algebra Level 3

What is the simplest form of the expression below?

x x + x x + x \sqrt{x-\sqrt{x+\sqrt{x-\sqrt{x+\sqrt{x-\cdots}}}}}

4 x + 3 1 4 \frac{\sqrt{4x+3}-1}{4} x 1 2 \frac{\sqrt{x}-1}{2} 5 x 3 1 3 \frac{\sqrt{5x-3}-1}{3} 4 x 3 1 2 \frac{\sqrt{4x-3}-1}{2}

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1 solution

Solving y = x x + y y=\sqrt{x-\sqrt{x+y}} gives four solutions: 1 2 ( 4 x 3 1 ) , \frac{1}{2} \left(-\sqrt{4 x-3}-1\right), 1 2 ( 4 x 3 1 ) , \frac{1}{2} \left(\sqrt{4 x-3}-1\right), 1 2 ( 1 4 x + 1 ) \frac{1}{2} \left(1-\sqrt{4 x+1}\right) and 1 2 ( 4 x + 1 + 1 ) \frac{1}{2} \left(\sqrt{4 x+1}+1\right) . Only one of those solutions is among the multiple choices offered as a solution. Therefore, that has to be the desired answer; even though I would have selected 1 2 ( 4 x + 1 + 1 ) \frac{1}{2} \left(\sqrt{4 x+1}+1\right) as the simplest form.

How do you know that the other 3 "solutions" are definitely wrong?

Pi Han Goh - 2 years, 1 month ago

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I did not say that they are wrong. When presented with a multiple choice situation, the answer is the form present among the choices.

A Former Brilliant Member - 2 years, 1 month ago

for x>0

1 2 ( 4 x 3 1 ) and 1 2 ( 1 4 x + 1 ) < 0 \dfrac12 (-\sqrt{4x-3}-1) \text{ and } \dfrac12(1-\sqrt{4x+1})<0

This will mean y<0 which is not possible as square root of a number denotes the positive root.

if y = 1 2 ( 1 + 4 x + 1 ) y 2 = x + y y = x + x + x + y=\dfrac12(1+\sqrt{4x+1})\\ y^2=x+y\\ \implies y=\sqrt{x+\sqrt{x+\sqrt{x+\cdots}}}

Anirudh Sreekumar - 2 years, 1 month ago

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The principal square root is the positive root. The square root operation has two roots, positive and negative.

A Former Brilliant Member - 2 years, 1 month ago

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