Question

If \[a - \dfrac{1}{a} = 5\] , find \[{a^2} + \dfrac{1}{{{a^2}}} - 3a + \dfrac{3}{a}\] ?

Answer 1

Hint: In this problem, we have to find the value of \[{a^2} + \dfrac{1}{{{a^2}}} - 3a + \dfrac{3}{a}\].
We will use the formula \[{\left( {a - b} \right)^2} = {a^2} + {b^2} - 2ab\] to solve the problem.
An algebraic expression consists of variables and constants combined together by some operators such as addition, subtraction, multiplication, division etc.

Complete step by step solution:
Now, we can find the \[{a^2} + \dfrac{1}{{{a^2}}} - 3a + \dfrac{3}{a}\] by the following steps
Given \[a - \dfrac{1}{a} = 5\] …………………………. (i)
Squaring both side we have ,
\[ \Rightarrow {\left( {a - \dfrac{1}{a}} \right)^2} = {\left( 5 \right)^2}\]
By using this algebraic formula \[{\left( {a - b} \right)^2} = {a^2} + {b^2} - 2ab\] to simplify the above, we get
\[ \Rightarrow {a^2} + \dfrac{1}{{{a^2}}} - 2 = 25\]
On simplifying by moving (-2) from LHS to RHS , we get
\[ \Rightarrow {a^2} + \dfrac{1}{{{a^2}}} = 27\] …………………………….. (ii)
Multiplying equation (i) by 3 and then subtracting from equation (ii) we have ,
\[ \Rightarrow \left( {{a^2} + \dfrac{1}{{{a^2}}}} \right) - 3\left( {a - \dfrac{1}{a}} \right) = 27 - 3 \times 5\].
On solving we get ,
\[ \Rightarrow {a^2} + \dfrac{1}{{{a^2}}} - 3a + \dfrac{3}{a} = 12\].
Hence \[12\] is the required solution.
As a result, If \[a - \dfrac{1}{a} = 5\] , find \[{a^2} + \dfrac{1}{{{a^2}}} - 3a + \dfrac{3}{a} = 12\].

Note:
We note that the constants are those whose value is fixed and does not change. While variables are those whose value changes. For example: x, y, z etc are variables. Whereas \[5,3,6\;\]etc are constants.
When algebraic expression is equated to zero it becomes an algebraic equation.
An algebraic expression can be easily solved by an algebraic formula. We can put into values and get the result without having to perform complex calculations.
Some important formulas are as below:
\[
  {\left( {a + b} \right)^2} = {a^2} + {b^2} + 2ab \\
  {\left( {a - b} \right)^2} = {a^2} + {b^2} - 2ab \\
 \]
For example if we want to find the value of \[105 \times 105\]. We will use the formula to find the value.
\[{105^2} = {\left( {100 + 5} \right)^2} = {100^2} + {5^2} + 2\left( {100} \right)\left( 5 \right)\]
On solving we get
\[{105^2} = {\left( {100 + 5} \right)^2} = 10000 + 25 + 1000 = 11025\].