If x is a real number and x + x^(1/2) = 1, which of the following is

What is wrong with this approach? 

x + x^1/2 = 1 --> x^1/2 = 1 - x

x^1/2 = 1 - x --> (x^1/2)^2 = (1 - x)^2

(x^1/2)^2 = (1 - x)^2 --> x = 1 + x^2 - 2x

x = 1 + x^2 - 2x --> x^2 - 3x + 1 = 0

Then using the quadratic equation we get the roots  3 + (5^1/2)/2 and 3 - (5^1/2)/2 and since x must be postive then Answer D. 3 + (5^1/2)/2?
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Several points:

1. The roots of \(x^2 - 3x + 1 = 0\) are \(\frac{3 - \sqrt{5}}{2}\approx 0.4\) and \(\frac{3 + \sqrt{5}}{2}\approx 2.6\). Both are positive, however, \(\frac{3 + \sqrt{5}}{2}\approx 2.6\) is not a valid root of \(x + \sqrt{x} = 1\): \(2.6 + \sqrt{2.6} > 1\). (When you squared you get that extra invalid root.)

2. We are asked to find the value of \(\sqrt{x}\), not x. So, if we substitute \(x=\frac{3 - \sqrt{5}}{2}\) into \(x + \sqrt{x} = 1\), we get \(\frac{3 - \sqrt{5}}{2} + \sqrt{x} = 1\), which yields \(\sqrt{x} = 1 - \left(\frac{3 - \sqrt{5}}{2}\right) = \frac{\sqrt{5} - 1}{2}\).

Hope it's clear.­

Because all mathematical explanations are already discussed so I am discussing a Purely Logical approach here

Given: x+√x = 1
Step-1: x = 1-√x


i.e. if √x = 0.8 then x = 0.64 but the sum of x and √x is greater than 1 lets take another approximation
if √x = 0.6 then x = 0.36 and the sum of x and √x in this case is 0.96 so very close to 1

so the correct option should be close to 0.6, Let's check options now,

A. \(\frac{1}{2}(\sqrt{5}-1)\) = (1/2)*(2.3-1) = 1.3/2 = 0.65 (Pretty close so possible answer)

B. \(\frac{1}{2}(\sqrt{5}+1)\) = (1/2)*(2.3+1) = 3.3/2 = 1.65 (Not possible)

C. \(\frac{1}{2}(\sqrt{5}-3)\) = (1/2)*(2.3-3) = -0.7/2 = Negative value (Not possible)

D. \(\frac{1}{2}(\sqrt{5}+3)\) = (1/2)*(2.3+3) = 5.3/2 = 2.65 (Not possible)

E. \(\frac{1}{2}(3-\sqrt{5})\) = (1/2)*(3-2.3) = 0.7/2 = 0.35 (not close to 0.6 hence Not possible)

Answer: Option A

P.S.: if you think that this approach doesn't work in real exam then I must say you are mistaken. This works wonderfully and flawlessly
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Essentially, the question says that one of the options (which is root(x)) added to its square (which is x) will give us 1. So the square roots will all get canceled off.

Four of the options have \(\frac{\sqrt{5}}{2}\)

Look for the option which upon squaring will give us \(- \frac{\sqrt{5}}{2}\)

Using \((a + b)^2 = a^2 + b^2 + 2ab\), we see that the root(5) term will come from 2ab.

Options (B) and (D) will give us positive root(5) terms upon squaring so they are not possible.

Option (A) will give us \(- \frac{2\sqrt{5}}{4} = - \frac{\sqrt{5}}{2}\) Looks good.

So simply check if it works.
\(x = (\frac{1}{2}(\sqrt{5}-1))^2 = \frac{1}{4}(5 + 1 - 2\sqrt{5}) = \frac{3}{2} - \frac{\sqrt{5}}{2} \)

\(x + \sqrt{x} = \frac{3}{2} - \frac{\sqrt{5}}{2} + \frac{1}{2}(\sqrt{5}-1) = 1\) - Valid


Answer (A)