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the equation: x + √x + √(x+2) + √(x^2 + 2x) = 3 has: 1. No solutions2. At least two solutions3. only positive solutions4. Infinite number of solutions |
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Answer» 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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