Question

A+1/a=4 and (a-1/a)=? factroise

Answer 1

Given;

$a + \frac{1}{a} = 4$

To find;

$a - \frac{1}{a} = ?$

Square both the sides of the given expression;

• using (a+b)² = a² + 2ab + b²

${(a + \frac{1}{a}) }^{2} = {4}^{2} \\ \implies {a}^{2} + \frac{1}{ {a}^{2} } + 2(a)( \frac{1}{a} ) =16 \\ \implies {a}^{2} + \frac{1}{ {a}^{2} } = 16 - 2 \\ \implies {a}^{2} + \frac{1}{{a}^{2} } = 14$

......(1)

Square the expression to be found both the sides ;

• using (a-b)² = a² + b² - 2ab

${(a - \frac{1}{a})}^{2} = {a}^{2} + \frac{1}{ {a}^{2} } - 2(a)( \frac{1}{a} )$

From (1)

$\implies {(a - \frac{1}{a})}^{2} = 14 - 2(a)( \frac{1}{a} ) \\ \implies {(a - \frac{1}{a})}^{2} = 14 - 2 \\ \implies {(a - \frac{1}{a})} = \sqrt{14 - 2} \\ \implies (a - \frac{1}{a}) = \sqrt{12}$

#answerwithquality

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