Question

a≠0 and a-1/a=4 find a2+1/a2

Answer 1

$a - \frac{1}{a} = 4 \: \\ we \: square \: both \: the \: sides \\ { (a - \frac{1}{a} ) }^{2} = {4}^{2} \\ = {a}^{2} + ( { \frac{1}{a} })^{2} + 2 = 16 \\ \\ = {a}^{2} + ( { \frac{1}{a} })^{2} = 16 - 2 = 14 \\ now \: in \\ 2a + \frac{1}{2a} = x \\ = 2(a + \frac{1}{a} ) = x \\ we \: squre \: both \: the \: sides \\ {(2(a + \frac{1}{a} ))}^{2} = {x}^{2} \\ = 4( {a}^{2} + {( \frac{1}{a}) }^{2} + 2) = {x}^{2} \\ = 4(14 + 2) = {x}^{2} \\ = 4 \times 16 = {x}^{2} \\ = {x}^{2} = 64 \\ = x = \sqrt{64} \\ = x = 8$

Answer 2

Correct Question :-

If a - 1/a = 4, then find the value of a² + 1/a²

Answer :-

a² + 1/a² = 18

Solution :-

a - 1/a = 4

Squaring on both sides

(a - 1/a)² = (4)²

⇒ (a - 1/a)² = 16

We know that

(x - y)² = x² + y² - 2xy

Here x = a, y = 1/a

By substituting the values

⇒ (a)² + (1/a)² - 2(a)(1/a) = 16

⇒ a² + 1²/a² - 2 = 16

⇒ a² + 1/a² - 2 = 16

Transpose - 2 to RHS ( - 2 becomes + 2)

⇒ a² + 1/a² = 16 + 2

⇒ a² + 1/a² = 18

Therefore the value of a² + 1/a² is 18.

Identity used :-

• (x - y)² = x² + y² - 2xy