Question

if x = ( 2 + root 3 ), find the value of x^2 + 1/x^2​

Answer 1

Answer:

14.Answer

Step-by-step explanation:

SEE THE IMAGE..

Answer 2

Question:-

• if x = ( 2 + root 3 ), find the value of x^2 + 1/x^2

Solution:-

$\small\sf\underline\green{given}$ ⟼x=2+√3

$\small\sf\underline\green{Need\:to\:find}$⟼x²+1/x²

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$x = 2 + \sqrt{ 3}$

$\frac{1}{x} = \frac{1}{2 + \sqrt{3} }$

After rationalizing the denominator we get as,

$\frac{1}{2 + \sqrt{3} } \times \frac{2 - \sqrt{3} }{2 - \sqrt{3} }$

$\frac{2 - \sqrt{3} }{4 - 3} = 2 - \sqrt{3}$

Now, to find the value of x²+1/x² we need to first find the value of x+1/x

$x + \frac{1}{x} = 2 + \sqrt{3} + 2 - \sqrt{3}$

$x + \frac{1}{x} = 2 + 2 = 4$

Now, we need to find the value of x²+1/x²

$(x + \frac{1}{x} ) {}^{2} = (4) {}^{2} = 16$

$(x + \frac{1}{x} ) {}^{2} = x {}^{2} + \frac{1}{ {x}^{2} } + 2 \times x \times \frac{1}{x}$

$x {}^{2} + \frac{1}{ {x}^{2} } + 2 = 16$

$\small\sf\red{ {x}^{2} + \frac{1}{ {x}^{2} } } = \small\sf\green{16 - 2 = 14}$

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Hence, the value of x²+1/x²= $\small\sf\underline\purple{14}$