Question

if х=9 - 4√5, find X²+1/X²
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Answer 1

Given:

• x = 9 - 4√5

To find:

x² - 1/x² = ?

Solution:

x = 9 - 4√5

Squaring on both sides,

x² = (9 - 4√5)²

Split using (a - b)² = a² + b² - 2ab

⇒ x² = (9)² + (4√5)² - 2(9)(4√5)

⇒ x² = 81 + 80 - 72√5

⇒ x² = 161 - 72√5

For finding the value of 1/x²

$\sf{\dfrac{1}{ {x}^{2} } = \dfrac{1}{161 - 72 \sqrt{5} } }$

Rationalize the Denominator

$\sf{ \dfrac{1}{ {x}^{2} } = \dfrac{1}{161 - 72 \sqrt{5} } \times \dfrac{161 + 72 \sqrt{5} }{161 + 72 \sqrt{5} } }$

Multiply using (a + b) (a - b) = a² - b²

$\sf{ \dfrac{1}{ {x}^{2} } = \dfrac{161 +72 \sqrt{5} }{(161)^{2} - (72 \sqrt{5}) ^{2} } }$

So

$\to \: \sf {\dfrac{1}{ {x}^{2} } = \dfrac{161 + 72 \sqrt{5} }{25921 - 25920} }$

Now we have

$\sf {\dfrac{1}{ {x}^{2} } = 161 + 72 \sqrt{5} }$

Adding x² and 1/x²

x² + 1/x²

= (161 - 72√5) + (161 + 72√5)

= 161 - 72√5 + 161 + 72√5

= 322

∴ x² + 1/x² = 322