Question

if,
$a - \frac{1}{a} = 5$
then what is the value of,
$(a + \frac{1}{a} {)}^{2}$

​

Answer 1

$(a + \dfrac{1}{a} {)}^{2} =25$

Step-by-step explanation:

Given:-

• $a - \dfrac{1}{a} = 5$

To Find:-

• $(a + \dfrac{1}{a} {)}^{2}$

Formula used:-

• (a - b)² = a² + b² - 2ab

• (a + b)² = a² + b² - 2ab

Solution:-

Firstly,

$a - \dfrac{1}{a} = 5$

Squaring both side,

$(a - \dfrac{1}{a} {)}^{2} = 5²$

Using Identity, (a - b)² = a² + b² - 2ab

$a² + \dfrac{1}{a²} - 2×a×\dfrac{1}{a}= 25$

$a² + \dfrac{1}{a²} = 25$

Now, Using identity (a + b)² = a² + b² - 2ab

$(a + \dfrac{1}{a} {)}^{2} = a² + \dfrac{1}{a²} +2×a×\dfrac{1}{a}$

$(a + \dfrac{1}{a} {)}^{2} = a² + \dfrac{1}{a²}$

Putting value,

${\boxed{(a + \dfrac{1}{a} {)}^{2} =25}}$

Hence, the value of ${\bold{(a + \dfrac{1}{a} {)}^{2}}}$ is 25.

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Answer 2

supposed to be the right answer in one method,

• If a-1/a=5 the a-1=5a
• -1=5a-a which is -1=4a which is a=-1/4 ————1
• a²-1/a² is (-1/4)² -1/(-1/4)² which is 1/16 -1/1/16.
• -15/16 / 1/16
• Applying another method from given condition in problem.
• Other right answer is a-1/a=5—Eq1
• So squaring both sides
• We get a² -2a/a +1/a² which is a² +1/a² -2=5×5
• Which further is a²+1/a²-2=25—Eq2
• Now adding 4 to both sides
• a²+1/a²-2+4=25+4
• Which is a²+2+1/a²=29
• Further square root of both sides
• √a²+2+1/a²=√29
• a+1/a=√29
• Now in problem the then asked what is a²-1/a² is (a+1/a)(a-1/a)
• a+1/a ×a-1/a
• Whereas in question a-1/a=5 and a+1/a=√29 is proved

So +-5√29 is Answer.