Math, asked by monabhat5454, 1 year ago

If x^2-4x+1=0 then find the value of x^3+1/x^3 by study iq

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Answered by KarupsK
8
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Answered by mysticd
2

 Given \:x^{2} - 4x + 1 = 0

 \implies x^{2} - 2\times x \times 2 + 2^{2} - 3 = 0

 \implies (x-2)^{2} = 3

 \implies x - 2 = \pm \sqrt{3}

 \implies x = 2 \pm \sqrt{3} \: --(1)

i )  \frac{1}{x} = \frac{1}{(2+\sqrt{3})}\\= \frac{(2-\sqrt{3})}{(2+\sqrt{3})(2-\sqrt{3})}\\= \frac{(2-\sqrt{3})}{2^{2}-(\sqrt{3})^{2}}\\= \frac{(2-\sqrt{3})}{4-3}\\= 2-\sqrt{3} \: --(2)

 x + \frac{1}{x} \\= 2 + \sqrt{3} + 2 - \sqrt{3} \\= 4 \: --(3)

 Now , x^{3} + \frac{1}{x^{3}} \\= \Big(x+\frac{1}{x}\Big)^{3} - 3\times x \times \frac{1}{x} \Big(x + \frac{1}{x}\Big) \\= 4^{3} - 3 \times 4 \\= 64 - 12 \\= 52

Therefore.,

 \red { Value \:of \: x^{3} + \frac{1}{x^{3}} } \green {= 52 }

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