Question

If ${a^2} + \dfrac{1}{{{a^2}}} = 786$ then find the value of $a - \dfrac{1}{a}$

Answer 1

Hint: We will use the given value and make changes accordingly to obtain the given value. We can see both the terms given are squares, thus we can apply completing the square method and find the required value

Complete step-by-step answer:
We have been given :
${a^2} + \dfrac{1}{{{a^2}}} = 786$
We need to find the value of $a - \dfrac{1}{a}$ , so we will make amendments so as to reach this value from the given value.
We can use the square method here.
We require ${\left( {a - \dfrac{1}{a}} \right)^2}$ and not ${\left( {a + \dfrac{1}{a}} \right)^2}$ so we will be using
${(a - b)^2} = {a^2} + {b^2} - 2ab$
Here, a = a and $b = \dfrac{1}{a}$ , we can add and subtract $2 \times a \times \dfrac{1}{a}$ to complete the square:
$
\Rightarrow {a^2} + \dfrac{1}{{{a^2}}} - 2 \times a \times \dfrac{1}{a} + 2 \times a \times \dfrac{1}{a} = 786 \\
  \Rightarrow {\left( {a - \dfrac{1}{a}} \right)^2} + 2 = 786 \\
\Rightarrow {\left( {a - \dfrac{1}{a}} \right)^2} = 786 - 2 \\
\Rightarrow {\left( {a - \dfrac{1}{a}} \right)^2} = 784 \\
 $
Square rooting both sides, we get:
\[
\Rightarrow \sqrt {{{\left( {a - \dfrac{1}{a}} \right)}^2}} = \sqrt {784} \\
\Rightarrow a - \dfrac{1}{a} = \pm 28 \\
 \]
Therefore, there are two possible values of $a - \dfrac{1}{a}$ that are: 28 and – 28

Note: When we perform modifications according to our needs, we always need to remember that the value of the main equation should not get changed i.e. if a value is added it should either be added on both sides or added or subtracted on the same side similarly in case of multiplication if a value gets multiplied on one side it should either get multiplied on both sides or multiplied or divided on the same side.
When the square root is taken the number obtained can be negative or positive, we consider both because a square is always positive (even for negative values) and before taking square root the number is a square and hence we take both the possibilities (positive and negative value) into consideration.