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0 votes
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in Complex number and Quadratic equations by (20 points)
reopened by
the equation: x + √x + √(x+2) + √(x^2 + 2x) = 3 has:
1. No solutions
2. At least two solutions
3. only positive solutions
4. Infinite number of solutions

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1 Answer

+1 vote
by (31.2k points)
edited by

Given equation is,

\(x\,+\sqrt{x}\,+\sqrt{x+2}\,+\sqrt{x^2+2x}\) = 3

\(\Rightarrow\) \(x\,+\sqrt{x}\,+\sqrt{x+2}\,+\sqrt{x^2+2x}-3=0\)

Let f(x) = \(x\,+\sqrt{x}\,+\sqrt{x+2}\,+\sqrt{x^2\,+\,2x}-3\)

Therefore, the domain of function f(x) is [0,).

f(o) = - 3 < 0

f(1) = 1 + 1 + \(\sqrt{3}\,+\sqrt{3}\)

= 2 + \(2\sqrt{3}\) > 0

Hence, at least one solution lies between 0 and 1.

f1(x) = 1 + \(\frac{1}{2\sqrt{x}}\) + \(\frac{1}{2\sqrt{x\,+\,2}}\) + \(\frac{2(x\,+\,1)}{2\sqrt{x^2\,+\,2x}}\)> 0 (\(\because\) \(x\in\) [0,∞)]

Hence, f(x) is increasing in its domain 

therefore, only one positive solution of given equation exist which lies between 0 and 1.

Hence, option (C) is correct

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