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Question

The quadratic equation whose one root is $$\cfrac{3+\sqrt{5}}{2-\sqrt{5}}$$ is


A
x2+22x+4=0
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B
x2+22x4=0
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C
x2+11x+8=0
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D
x2+11x8=0
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Solution

The correct option is B $${x}^{2}+22x-4=0$$
One root $$\displaystyle =\frac{3+\sqrt{5}}{2-\sqrt{5}}\frac{(2+\sqrt{5})}{(2+\sqrt{5})}$$
                $$\displaystyle =\frac{6+3\sqrt{5}+2\sqrt{5}+5}{4-5}$$
                $$=-(11+5\sqrt{5})=-11-5\sqrt{5}$$
Other root $$\Rightarrow -11+5\sqrt{5}$$
Equation $$\Rightarrow x^{2}-(-11-5\sqrt{5}-11+5\sqrt{5})x+(-11-5\sqrt{5})(-11+5\sqrt{5})=0$$
$$x^{2}+22x+(121-125)=0$$
$$x^{2}+22x-4=0$$

Mathematics

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