If a - 1-a - 17-4- Find a - 1-a

Step 1: Squaring both sides in the given equationGiven equation: a + 1/a = 17/4On Squaring both sides, we get,(a+1/a)^2 = (17/4)^2a^2 + 1/a^2 + 2 = 289/16 ( A ...

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Standard VIII

Mathematics

Addition and Subtraction of Irrational Numbers

If a + 1/a = ...

Question

If $a + \frac{1}{a} = \frac{17}{4}$ ; Find ( $a - \frac{1}{a}$ )

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Solution

Step 1: Squaring both sides in the given equation
Given equation: $a + \frac{1}{a} = \frac{17}{4}$
On Squaring both sides, we get,
${(a + \frac{1}{a})}^{2} = {(\frac{17}{4})}^{2}$
$a^{2} + \frac{1}{a^{2}} + 2 = \frac{289}{16}$ ( As ${(a + b)}^{2} = a^{2} + b^{2} + 2 a b$ )
$a^{2} + \frac{1}{a^{2}} = \frac{289}{16} - 2$ ( Transposing $2$ from left side to right side and while doing so addition becomes subtraction )
$a^{2} + \frac{1}{a^{2}} = \frac{289 - 32}{16}$ ( Taking LCM of Right side part and solving further )
$a^{2} + \frac{1}{a^{2}} = \frac{257}{16}$ ----------- (i)
Step 2: Now, Squaring $a - \frac{1}{a}$
${(a - \frac{1}{a})}^{2} = a^{2} + \frac{1}{a^{2}} - 2$ ( As ${(a - b)}^{2} = a^{2} + b^{2} - 2 a b$ )
${(a - \frac{1}{a})}^{2} = \frac{257}{16} - 2$ ( Substituting the value of $a^{2} + \frac{1}{a^{2}}$ from (i) )
${(a - \frac{1}{a})}^{2} = \frac{257 - 32}{16}$ ( Taking LCM of Right side part and solving further )
${(a - \frac{1}{a})}^{2} = \frac{225}{16}$
Taking square root on both sides,
$\sqrt{{(a - \frac{1}{a})}^{2}} = \sqrt{\frac{225}{16}}$
$(a - \frac{1}{a}) = \frac{15}{4}$
Hence, the value of $(a - \frac{1}{a})$ is $\frac{15}{4}$ .

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If a+1a=174; Find (a-1a)

If $a + \frac{1}{a} = \frac{17}{4}$ ; Find ( $a - \frac{1}{a}$ )