Algebra

high importance~3 Q in Tier 143 formulas⚡ 21 shortcuts6 subtopics

Overview Learn (6)Formulas (43)Examples (4)Practice (78)

Identities, $x+\frac1x$ expressions, $a^3+b^3+c^3-3abc$ , surds, linear equations and simple max–min. Algebra gives 2–4 marks-rich questions in every CGL shift, and almost all of them fall to a handful of identities plus value-putting — the highest return per hour of study in advanced maths.

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Learn — 6 subtopics

One page per subtopic: detailed notes, every question type, formulas, tricks and practice sets.

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Formula sheet

Every formula on one printable page, grouped by subtopic.

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Solved examples

4 exam-level questions worked step by step.

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Practice & test

78 questions — untimed practice or a timed test with analysis.

Track record in the exam

avg 2.0 Q / shift2024: 2–3 Q2025: 2 Q

Questions per shift in recent SSC CGL papers.

Test difficulty mix (78 questions)

26 easy41 medium11 hard

Question patterns exams keep repeating

Taken from previous-year papers. If a pattern is marked "very common", expect to see it in your exam.

$x+\frac1x=k$ → higher powers

very common

Spot it:

Given $x+\frac1x$ , $x-\frac1x$ or a quadratic with equal end coefficients; asked for $x^3+\frac1{x^3}$ , $x^4+\frac1{x^4}$ , $x^5+\frac1{x^5}$ .

How to solve: Climb the ladder: square and subtract 2; cube with $k^3-3k$ ; fifth rung $= (k^2-2)(k^3-3k)-k$ . Recognise special values ( $1,\ -1,\ \sqrt3$ ) for cyclic exponents.

Example: If $x+\frac1x=6$ , then $x^3+\frac1{x^3}$ is:

$6^3 - 3\times6 = 216 - 18 = 198$ .

Learn this in “$x+\frac{1}{x}$ type expressions” →

Given $a\pm b$ and $ab$, find $a^3\pm b^3$ / $a^4+b^4$

very common

Spot it:

Two conditions on two variables and a symmetric expression to evaluate.

How to solve: Never solve for $a,b$ . Build upward: $(a\pm b)^2 \mp 2ab$ for squares, $(a\pm b)^3 \mp 3ab(a\pm b)$ for cubes, $(a^2+b^2)^2-2a^2b^2$ for fourth powers.

Example: If $a+b=7$ and $ab=12$ , then $a^3+b^3$ is:

$343 - 3\times12\times7 = 343 - 252 = 91$ .

Learn this in “Basic algebraic identities” →

Conjugate surd $x=p+\sqrt q$

very common

Spot it:

$x=2+\sqrt3$ , $3-2\sqrt2$ , $\frac{1}{\sqrt5-2}$ etc., asked for $x^2+\frac1{x^2}$ or a polynomial in $x$ .

How to solve: Check $p^2-q=1$ → $\frac1x=p-\sqrt q$ ; then $x+\frac1x = 2p$ and run the ladder. Otherwise reduce the polynomial modulo the minimal quadratic.

Example: If $x=2+\sqrt3$ , then $x^2+\frac1{x^2}$ is:

$\frac1x = 2-\sqrt3$ → $x+\frac1x = 4$ → $x^2+\frac1{x^2} = 14$ .

Learn this in “Surds: rationalisation and square roots of surds” →

$a^3+b^3+c^3-3abc$ / zero-sum cubes

common

Spot it:

Three variables with $a+b+c$ given, or expressions like $(x-y)^3+(y-z)^3+(z-x)^3$ .

How to solve: Factor form $s(s^2-3P)$ ; zero sum → cube-sum $= 3abc$ ; power sum $s^3 - 3sP + 3R$ when $abc$ is in play.

Example: If $a+b+c=6$ and $ab+bc+ca=11$ , then $a^3+b^3+c^3-3abc$ is:

$6\times(36-33) = 18$ .

Learn this in “$a^3+b^3+c^3-3abc$ and conditional identities” →

Sum of squares equal to zero

common

Spot it:

An equation like $x^2+y^2+z^2-2x+4y-6z+14=0$ or $a^2+b^2+c^2=ab+bc+ca$ .

How to solve: Complete squares so the constants cancel; each square is zero, giving exact values. The pairwise form forces $a=b=c$ .

Example: If $x^2+y^2-8x+6y+25=0$ , then $x-y$ is:

$(x-4)^2+(y+3)^2=0$ → $x=4$ , $y=-3$ → $x-y=7$ .

Learn this in “Basic algebraic identities” →

Big-number / decimal simplification

common

Spot it:

Fractions with cubes over quadratics of the same two numbers, e.g. $\frac{p^3+q^3}{p^2-pq+q^2}$ .

How to solve: Match the identity using the sign of the denominator's middle term; answer is $p\pm q$ .

Example: The value of $\dfrac{4.7^3+2.3^3}{4.7^2-4.7\times2.3+2.3^2}$ is:

$p^2-pq+q^2$ under $p^3+q^3$ → the fraction $= 4.7+2.3 = 7$ .

Learn this in “Basic algebraic identities” →

Linear equations, graphs, remainder

occasional

Spot it:

Condition for unique/no/infinite solutions, area of triangle with axes, remainder or factor of a polynomial.

How to solve: Ratios of coefficients; area $= \frac{c^2}{2ab}$ from the intercepts; remainder $= p(\alpha)$ by direct substitution.

Example: The remainder when $x^3-3x^2+4x-5$ is divided by $(x-2)$ is:

$p(2) = 8-12+8-5 = -1$ .

Learn this in “Linear equations, graphs and polynomials” →

Maximum / minimum value

occasional

Spot it:

'Least value of $4x+\frac9x$ ', 'greatest value of $7+4x-x^2$ '.

How to solve: AM ≥ GM for $ax+\frac bx$ (value $2\sqrt{ab}$ ); complete the square for quadratics (vertex $c-\frac{b^2}{4a}$ ).

Example: The greatest value of $7+4x-x^2$ is:

$-(x-2)^2+11 \le 11$ → greatest value 11 at $x=2$ .

Learn this in “Maxima and minima (AM ≥ GM, quadratics)” →

Your next step

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