Algebra Decoded: Identities and Shortcuts for SSC-CGL
Build algebra intuition for SSC-CGL with must-know identities, the a+1/a family of tricks, and a step-by-step method to crack value-finding questions fast.
By the PadhoDost Team · 📖 8 min read · Updated 4 August 2026
Part of SSC CGL & Banking prep🧠 Algebra is a balance scale
Think of an equation as an old-style weighing balance in a sabzi mandi. Whatever you do to one pan, you must do to the other, or it tips. Add 5 grams to the left? Add 5 to the right. That single rule - keep both sides equal - is the entire soul of algebra. Everything else is just clever ways to rearrange the weights so the unknown x sits alone on one pan.
In SSC-CGL, algebra is rarely about solving for x the slow way. It is about recognising a pattern - an identity - and jumping straight to the answer. Master a handful of identities and most questions collapse into one or two lines.
The identities you must know cold
| Identity | Expansion |
|---|---|
| (a + b)^2 | a^2 + 2ab + b^2 |
| (a - b)^2 | a^2 - 2ab + b^2 |
| a^2 - b^2 | (a + b)(a - b) |
| (a + b)^3 | a^3 + b^3 + 3ab(a + b) |
| (a - b)^3 | a^3 - b^3 - 3ab(a - b) |
| a^3 + b^3 | (a + b)(a^2 - ab + b^2) |
| a^3 - b^3 | (a - b)(a^2 + ab + b^2) |
| a^3 + b^3 + c^3 - 3abc | (a + b + c)(a^2 + b^2 + c^2 - ab - bc - ca) |
The a + 1/a family (SSC's favourite)
A whole cluster of questions gives you a + 1/a and asks for a^2 + 1/a^2 or a^3 + 1/a^3. These are built from the square and cube identities, because the cross-term neatly becomes 2 or 3.
How to attack an 'a + 1/a' question
- 1Read what you are given: is it a + 1/a or a - 1/a? The sign changes the -2 to +2.
- 2To climb from power 1 to power 2, square the given value and subtract 2 (for the plus version).
- 3To climb to power 3, cube the given value and subtract 3 times the given value.
- 4Never solve for a itself - stay inside the identity and just plug numbers.
- 5Double-check the sign of the constant: (a - 1/a)^2 = a^2 + 1/a^2 - 2, so a^2 + 1/a^2 = (a - 1/a)^2 + 2.
📝 Worked example
Given: a + 1/a = 4. Find a^3 + 1/a^3.
Use a^3 + 1/a^3 = (a + 1/a)^3 - 3(a + 1/a).
= 4^3 - 3 x 4
= 64 - 12
= 52.
Quick cross-check via a^2 + 1/a^2 = 4^2 - 2 = 14, all consistent.
Exam-day essentials
- ✓Learn the 8 core identities in the table so you recognise them at a glance.
- ✓a + b + c = 0 implies a^3 + b^3 + c^3 = 3abc - a huge time-saver.
- ✓For a + 1/a: square-minus-2 for the square, cube-minus-3-times for the cube.
- ✓For a - 1/a: square-PLUS-2 for the square; the sign flips only on the even step.
- ✓Substitute values into identities; almost never solve the underlying quadratic.
⚡ Quick check
If a - 1/a = 3, what is the value of a^2 + 1/a^2?
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