Question

PLEASE SOLVE THIS QUESTION​

Answer 1

Answer:

The answer is 14

Step-by-step explanation:

Given $x=2+\sqrt 3$

$\Rightarrow \frac{1}{x}=\frac{1}{2+\sqrt 3}\times \frac{2-\sqrt 3}{2-\sqrt 3}$, rationalize the denominator

$\Rightarrow \frac{1}{x}=\frac{2-\sqrt 3}{4-3}=2-\sqrt 3$

So $x+\frac{1}{x}=2+\sqrt 3+2-\sqrt 3=4$

Now square both sides

$\Rightarrow (x+\frac{1}{x})^2=4^2\\\Rightarrow x^2+\frac{1}{x^2}+2\times x\times \frac{1}{x}=16\\\Rightarrow x^2+\frac{1}{x^2}+2=16\\\therefore x^2+\frac{1}{x^2}=16-2=14$

Answer 2

Question:

If x = 2 + √3, find the value of x² + 1/x²

Answer:

14

Step-by-step explanation:

Given: x = 2 + √3 we need to find out the value of x² + 1/x².

Now,

→ x = 2 + √3

Do squaring on the both sides,

→ x² = (2 + √3)²

→ x² = 4 + 3 + 4√3

Used formula: (a + b)² = a² + b² + 2ab

→ x² = 7 + 4√3 -------------- (eq 1)

Similarly,

→ 1/x² = 1/(7 + 4√3)

Rationalise the denominator. While rationalising we use the opposite sign as that of the given sign in denominator and then multiply & divide it.

→ 1/x² = 1/(7 + 4√3) × (7 - 4√3)/(7 - 4√3)

→ 1/x² = (7 - 4√3)/[(7 + 4√3)(7 - 4√3)]

Used formula: (a + b) (a - b) = a² - b²

→ 1/x² = (7 - 4√3)/(49 - 48)

→ 1/x² = 7 - 4√3

Now,

→ x² + 1/x² = 7 + 4√3 + 7 - 4√3

→ x² + 1/x² = 14

Therefore, the value of x² + 1/x² is 14.