Tips & Tricks for Algebra

Last Updated : 24 Jun, 2026

Algebra is key to aptitude tests, helping solve problems quickly and efficiently. By mastering simple tricks like simplifying equations, spotting patterns, and using shortcuts, you can save time and boost accuracy.

Below are some of the tricks and shortcut formulas to quickly solve algebraic equations.

Trick 1: Simplifying Powers of x ± 1/x:

Given: x + 1/x = k, find x2 + 1/x2.

Step 1: Square both sides

  • (x + 1/x)2 = k2

Step 2: Expand

  • x2 + 1/x2 + 2 = k2

Step 3: Rearrange

  • x2 + 1/x2 = k2- 2

Note: similarly for x - 1/x = k the value will be K2 + 2

Trick 2: Trick for Simplifying Higher Powers of x + 1/x and x − 1/x

The higher powers of x + 1/x2 can be solved quickly by applying the formulas below:

  • x2 + 1/x2 = a2 - 2
  • x3 + 1/x3 = a3 - 3a
  • x4 + 1/x4 = a4 - 4a2 + 2
  • x5 + 1/x5 = a5 - 5a3 + 5a
  • x6 + 1/x6 = a6 - 6a4 + 9a2 - 2

Similarly for x - 1/x = a

  • x2 + 1/x2 = a2 + 2
  • x3 + 1/x3 = a3 + 3a
  • x4 + 1/x4 = a4 + 4a2 + 2
  • x5 + 1/x5 = a5 + 5a3 + 5a
  • x6 + 1/x6 = a6 + 6a4 + 9a2 + 2

Trick 3: Simplifying Powers of x + 1/x = 2

  • If x + 1/x = 2 , then for any positive integer n, the value of any equation xm + 1/xn will also be equal to 2.
  • If x + 1/x = 2 , then for any positive integer n, the value of any equation xn + 1/xn will also be equal to 2.
  • If x + 1/x = 2 , then for any positive integer n, the value of any equation xn - 1/xn will also be equal to 0.
  • If x + 1/x = 2 , then for any positive integer n, the value of any equation xm - 1/xn be equal to 0.

Trick 4: Quick Formulas for x − 1/x​ and x + 1/x​ Relations

This trick helps you find the value of x − 1/x or x + 1/x​ using a simple formula when the other is given.

The value of x - 1/x can be easily be found by applying the formula √m2 - 4

Similarly for the second equation

The value of x + 1/x can be easily be found by applying the formula √n2 + 4

Solved Examples

Example 1: If x + 1/x = 18 then find the value of x - 1/x..

Applying the formula: √m2 - 4

= √182 - 4

= √324 - 4

= ±√320 or ±8√5

Example 2: If x - 1/x = 11 then find the value of x + 1/x.

Applying the formula: √n2 + 4

= √112 + 4

= √121 + 4

= ±√125 or ±5√5

Example 3: If (x + 1/x= 5), find the value of (x2 + 1+1/x2).

Given, x +1/x= 5

Using the shortcut formula, x^2 + \frac{1}{x^2} = a^2 - 2 where (a = x + \frac{1}{x}).

Substituting (a = 5), x^2 + \frac{1}{x^2}\\[3pts] 5^2 - 2 = 25 - 2 = 23

Practice Problems

Problem 1 : If (x + \frac{1}{x} = 7), find the value of x^2 + \frac{1}{x^2}

Problem 2 : If (x + \frac{1}{x} = 4), find the value of x^4 + \frac{1}{x^4}.

Problem 3 : If (x - \frac{1}{x} = 5), find the value of x^3 + \frac{1}{x^3}.

Problem 5 : If (x + \frac{1}{x} = 6), find the value of x - \frac{1}{x}.

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