Question 106

If $$x^2 -\sqrt{7}x +1 =0$$, then $$(x^3 + x^{-3}) = ?$$

Solution

$$x^2 -\sqrt 7 x +1 = 0 $$

$$\Rightarrow x^2 + 1 = \sqrt 7 x $$

$$\Rightarrow \dfrac {x^2 +1}{x} = \sqrt 7 $$

$$\Rightarrow x + \dfrac{1}{x} = \sqrt 7 $$

so $$ x^3 + x^-3 = x^3 + \dfrac{1}{x^3}$$

$$\Rightarrow (x + \dfrac{1}{x})^3 - 3x\dfrac{1}{x}(x+\dfrac{1}{x})$$

$$\Rightarrow (\sqrt7)^3 - 3 \sqrt7 $$

$$\Rightarrow 7\sqrt 7 - 2\sqrt 7 $$

$$\Rightarrow 4 \sqrt7 $$ Ans

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