Question

If a2+1/a2=14. find a+1/a ii) a-1/a. iii) a2-1/a2​

Answer 1

SOLUTION

GIVEN

$\displaystyle\sf{ {a}^{2} + \frac{1}{ {a}^{2} } = 14}$

TO DETERMINE

The value of

$\displaystyle\sf{ i) \: \: {a}^{} + \frac{1}{ {a}^{} } }$

$\displaystyle\sf{ ii) \: {a}^{} - \frac{1}{ {a}^{} } }$

$\displaystyle\sf{ iii) \: \: {a}^{2} - \frac{1}{ {a}^{2} } }$

EVALUATION

Here it is given that

$\displaystyle\sf{ {a}^{2} + \frac{1}{ {a}^{2} } = 14}$

$\displaystyle\sf{ \implies \: { \bigg(a + \frac{1}{a} \bigg)}^{2} - 2.a. \frac{1}{a} = 14}$

$\displaystyle\sf{ \implies \: { \bigg(a + \frac{1}{a} \bigg)}^{2} - 2 = 14}$

$\displaystyle\sf{ \implies \: { \bigg(a + \frac{1}{a} \bigg)}^{2} = 16}$

$\displaystyle\sf{ \implies \: { \bigg(a + \frac{1}{a} \bigg)}^{} =4}$

$\displaystyle\sf{ i) \: \: \: { \bigg(a + \frac{1}{a} \bigg)}^{} = 4}$

ii)

$\displaystyle\sf{ {a}^{2} + \frac{1}{ {a}^{2} } = 14}$

$\displaystyle\sf{ \implies \: { \bigg(a - \frac{1}{a} \bigg)}^{2} + 2.a. \frac{1}{a} = 14}$

$\displaystyle\sf{ \implies \: { \bigg(a - \frac{1}{a} \bigg)}^{2} + 2 = 14}$

$\displaystyle\sf{ \implies \: { \bigg(a - \frac{1}{a} \bigg)}^{2} = 12}$

$\displaystyle\sf{ \implies \: { \bigg(a - \frac{1}{a} \bigg)}^{} =2 \sqrt{3} }$

iii)

$\displaystyle\sf{ {a}^{2} - \frac{1}{ {a}^{2} } }$

$\displaystyle\sf{ = \: { \bigg(a + \frac{1}{a} \bigg)}^{} { \bigg(a - \frac{1}{a} \bigg)}^{} }$

$\displaystyle\sf{ = 4 \times 2 \sqrt{3} }$

$\displaystyle\sf{ = 8 \sqrt{3} }$

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