Variable
A symbol such as \(x\) whose value may vary or be unknown.
12 SSC CGL · Mathematics
Algebra
Handwritten-style notes for SSC CGL Tier I and Tier II.
01MAP IT!
The Big Picture — Graphic Mind Map
Algebra translates patterns into symbols, transforms them without changing value, and solves for the unknown.
ALGEBRA
LANGUAGE
& LOGIC
LANGUAGE
& LOGIC
1. Expressionsterms, coefficients, degree and evaluation
2. Powerslaws of exponents and elementary surds
3. Identitiesrecognize and expand standard patterns
4. Factorizationreverse expansion into useful products
5. Equationsone variable, two variables and word models
6. Graphsaxes, intercepts, slope and intersections
Algebra is a reversible machine
Expansion opens brackets; factorization closes them. Solving preserves equality while isolating the unknown. Substitution checks that the output truly fits.
\[\text{translate}\to\text{simplify}\to\text{solve}\to\text{verify}\]
\[\text{identity: true for all allowed values}\qquad\text{equation: true only for its solution set}\]
02ZERO TO EXPERT
The Foundation (Basics) — Complete Concept Build
Learn the language of terms and operations, then build identities, equations, graphs and surds.
AAlgebraic language
Constant
A fixed number such as \(7\), \(-3\) or \(\pi\).
Term
A product of numbers and variables separated from other terms by addition or subtraction.
\[5x^2y\text{ is one term}\]
Coefficient
The numerical multiplier of a variable part. In \(-7x^3\), the coefficient is \(-7\).
Expression
A meaningful algebraic combination without an equality sign, such as \(2x+3\).
Equation
A statement that two expressions are equal, such as \(2x+3=11\).
Like terms
Terms with identical variable parts and exponents.
\[3x^2+5x^2=8x^2\]
Unlike terms
\[x+x^2\text{ cannot be combined}\]
Degree
For a term, add variable exponents. For a polynomial, take the highest term degree.
\[\deg(4x^2y^3)=5\]
Only identical variable labels combine
Coefficients add or subtract; variable parts stay unchanged. Multiplication, however, combines exponents by the exponent laws.
BPolynomial classification and evaluation
| Classification | Meaning | Example |
|---|---|---|
| Monomial | One non-zero term | \(5x^3\) |
| Binomial | Two unlike terms | \(x+4\) |
| Trinomial | Three unlike terms | \(x^2+3x+2\) |
| Linear | Degree \(1\) | \(2x-5\) |
| Quadratic | Degree \(2\) | \(x^2-9\) |
| Cubic | Degree \(3\) | \(x^3+1\) |
Substitution
Replace every occurrence of a variable with its value, using brackets for negatives.
\[x=-2:\quad x^2=(-2)^2=4\]
Order of operations
Brackets, powers/roots, multiplication/division, then addition/subtraction.
Domain restriction
Values making a denominator zero or an even-root radicand negative are excluded in real algebra.
\[\frac1{x-2}:\quad x\ne2\]
CLaws of exponents
Same base: multiply
\[a^m a^n=a^{m+n}\]
Same base: divide
\[\frac{a^m}{a^n}=a^{m-n}\quad(a\ne0)\]
Power of a power
\[(a^m)^n=a^{mn}\]
Power of product/quotient
\[(ab)^n=a^nb^n,\qquad\left(\frac ab\right)^n=\frac{a^n}{b^n}\]
Zero and negative powers
\[a^0=1,\qquad a^{-n}=\frac1{a^n}\quad(a\ne0)\]
Fractional power
\[a^{m/n}=\sqrt[n]{a^m}\]
For even \(n\), use a non-negative real radicand.
Exponent rules count repeated factors
Multiplying same-base powers joins factor strings, so exponents add. Raising a power repeats the whole string, so exponents multiply.
\[\underbrace{a\cdots a}_{m}\underbrace{a\cdots a}_{n}=\underbrace{a\cdots a}_{m+n}\]
DEssential square identities
Square of a sum
\[\boxed{(a+b)^2=a^2+2ab+b^2}\]
Square of a difference
\[\boxed{(a-b)^2=a^2-2ab+b^2}\]
Difference of squares
\[\boxed{a^2-b^2=(a-b)(a+b)}\]
Two linear factors
\[(x+a)(x+b)=x^2+(a+b)x+ab\]
Three-term square
\[(a+b+c)^2=a^2+b^2+c^2+2ab+2bc+2ca\]
Useful sum
\[(a+b)^2+(a-b)^2=2(a^2+b^2)\]
The middle term appears twice
A square of side \(a+b\) contains one \(a^2\) square, one \(b^2\) square and two \(ab\) rectangles.
ECubic identities and three-variable identity
Cube of sum
\[(a+b)^3=a^3+3a^2b+3ab^2+b^3\]
Cube of difference
\[(a-b)^3=a^3-3a^2b+3ab^2-b^3\]
Sum of cubes
\[a^3+b^3=(a+b)(a^2-ab+b^2)\]
Difference of cubes
\[a^3-b^3=(a-b)(a^2+ab+b^2)\]
Three-variable factor
\[a^3+b^3+c^3-3abc=(a+b+c)(a^2+b^2+c^2-ab-bc-ca)\]
If the sum is zero
\[a+b+c=0\Rightarrow a^3+b^3+c^3=3abc\]
FIdentity-based evaluation
Given sum and product
\[a^2+b^2=(a+b)^2-2ab\]
\[(a-b)^2=(a+b)^2-4ab\]
Reciprocal square
\[\left(x+\frac1x\right)^2=x^2+\frac1{x^2}+2\]
Reciprocal cube
\[x^3+\frac1{x^3}=\left(x+\frac1x\right)^3-3\left(x+\frac1x\right)\]
Difference reciprocal
\[\left(x-\frac1x\right)^2=x^2+\frac1{x^2}-2\]
Sign matters
From a known square, taking a square root may produce two signs:
\[y^2=k\Rightarrow y=\pm\sqrt{k}\]
Use the given pattern as a key
Do not calculate unknown values individually if the required expression can be generated directly by squaring, cubing or combining the given expression.
\[\text{given pattern}\xrightarrow{\text{identity}}\text{target pattern}\]
GFactorization — reverse the expansion
Common factor
\[ax+ay=a(x+y)\]
Grouping
\[ax+ay+bx+by=(a+b)(x+y)\]
Difference of squares
\[x^2-a^2=(x-a)(x+a)\]
Monic trinomial
For \(x^2+px+q\), find \(m,n\) with \(m+n=p\) and \(mn=q\):
\[x^2+px+q=(x+m)(x+n)\]
Non-monic trinomial
For \(ax^2+bx+c\), split the middle term using numbers whose sum is \(b\) and product is \(ac\), then group.
Cubes
Use the sum/difference-of-cubes patterns. The second bracket signs follow “same, opposite, always positive square.”
Factorization asks for sum and product
The pair must add to the middle coefficient and multiply to the constant contribution. Always multiply the factors back to verify.
HAlgebraic fractions
Factor before cancelling
\[\frac{x^2-a^2}{x-a}=x+a\quad(x\ne a)\]
Cancel factors, not terms
\[\frac{x+2}{x}\ne2\]
Common denominator
\[\frac ab+\frac cd=\frac{ad+bc}{bd}\quad(bd\ne0)\]
Restrictions survive simplification
Even if a factor cancels, the original denominator restriction remains part of the domain.
Complex fraction
Multiply numerator and denominator by the LCM of smaller denominators; never invert only one term.
ILinear equation in one variable
Standard form
\[ax+b=c\quad(a\ne0)\Rightarrow x=\frac{c-b}{a}\]
Balance principle
Perform the same valid operation on both sides. This preserves the solution set.
Fractions
Multiply every term on both sides by the LCM of denominators, while recording denominator restrictions.
Variables on both sides
\[ax+b=cx+d\Rightarrow(a-c)x=d-b\]
No solution
\[0x=k\quad(k\ne0)\]
Infinitely many solutions
\[0x=0\]
The original sides simplify to the same expression.
An equation is a balanced scale
Whatever you add, subtract, multiply or divide on one side must be done to the other. Division by zero is never allowed.
\[L=R\Rightarrow L+k=R+k\]
JTranslating word problems
| Language | Algebraic model | Watch point |
|---|---|---|
| A number | Let it be \(x\) | Define the variable with its unit |
| Consecutive integers | \(x,x+1,x+2\) | Even/odd consecutive values differ by \(2\) |
| Two-digit number | \(10a+b\) | Digits satisfy \(0\le a,b\le9\) and \(a\ne0\) |
| Reversed two-digit number | \(10b+a\) | Difference equals \(9(a-b)\) |
| Age after/before \(t\) years | \(x+t\) / \(x-t\) | All persons move by the same number of years |
| Fraction numerator/denominator changed | \((a\pm k)/(b\pm k)\) | Translate each change separately |
| Ratio \(m:n\) | Values \(mk,nk\) | Use a common multiplier \(k\) |
KTwo simultaneous linear equations
Standard pair
\[a_1x+b_1y=c_1,\qquad a_2x+b_2y=c_2\]
Substitution
Express one variable from one equation, substitute into the other, then back-substitute.
Elimination
Scale equations so one variable has equal or opposite coefficients; add or subtract to eliminate it.
Unique solution
\[\Delta=a_1b_2-a_2b_1\ne0\]
No solution
\[\frac{a_1}{a_2}=\frac{b_1}{b_2}\ne\frac{c_1}{c_2}\]
Infinitely many
\[\frac{a_1}{a_2}=\frac{b_1}{b_2}=\frac{c_1}{c_2}\]
The common solution is the intersection
Algebraic elimination and graphical intersection describe the same event. Parallel distinct lines have no common point; coincident lines share infinitely many.
LCoordinate axes and graphs of linear equations
Coordinate point
\((x,y)\) means move \(x\) units horizontally, then \(y\) units vertically.
Quadrant signs
\[I:(+,+),\ II:(-,+),\ III:(-,-),\ IV:(+,-)\]
Line form
\[ax+by=c\]
Every point satisfying the equation lies on its straight-line graph.
Intercepts
\[x\text{-intercept}=\frac ca,\qquad y\text{-intercept}=\frac cb\]
Set the other coordinate to zero.
Slope
\[y=-\frac abx+\frac cb\Rightarrow m=-\frac ab\]
Special lines
\[x=k\text{ is vertical},\qquad y=k\text{ is horizontal}\]
Plot two points to draw a line
The intercept method is fastest when both intercepts exist. Otherwise choose two convenient values of one coordinate and compute the other.
\[x=0\Rightarrow y=c/b,\qquad y=0\Rightarrow x=c/a\]
MElementary surds
Principal square root
\[\sqrt{a^2}=|a|\]
The radical symbol denotes the non-negative square root.
Simplify perfect-square factors
\[\sqrt{ab}=\sqrt a\sqrt b\quad(a,b\ge0)\]
\[\sqrt{72}=6\sqrt2\]
Like surds
\[p\sqrt m+q\sqrt m=(p+q)\sqrt m\]
Do not split sums
\[\sqrt{a+b}\ne\sqrt a+\sqrt b\]
Conjugates
\[(a+\sqrt b)(a-\sqrt b)=a^2-b\]
Rationalize
\[\frac1{a+\sqrt b}=\frac{a-\sqrt b}{a^2-b}\]
Require a non-zero original and final denominator.
Conjugates cancel the radical cross-terms
Keep every term and flip only the middle sign. Their product becomes a difference of squares.
NFormula bank and verification
| Goal | Fast pattern | Verification |
|---|---|---|
| Expand square | First square, twice product, second square | Substitute simple values such as \(a=1,b=1\) |
| Factor quadratic | Find required sum and product | Multiply factors back |
| Solve equation | Preserve equality and isolate variable | Substitute into original equation |
| Solve two equations | Eliminate or substitute | Check both equations |
| Draw line | Plot two valid points | Both coordinates satisfy the equation |
| Simplify surd | Extract perfect-square factors | Square positive result where appropriate |
03SAVE TIME
Short Tricks & Magic Formulas
Recognize the pattern before expanding; often the shortest algebra does less arithmetic.
1Identity speed bank
Near-base square
\[(a\pm b)^2=a^2\pm2ab+b^2\]
Product around a centre
\[(x-a)(x+a)=x^2-a^2\]
Consecutive offsets
\[(x+a)(x+b)=x^2+(a+b)x+ab\]
2Given-sum/given-product tricks
Squares sum
\[a^2+b^2=(a+b)^2-2ab\]
Difference magnitude
\[|a-b|=\sqrt{(a+b)^2-4ab}\]
Cubes sum
\[a^3+b^3=(a+b)^3-3ab(a+b)\]
3Factor and cancel safely
GCF first
Remove the greatest common factor before attempting grouping or identities.
Quadratic pair
For \(x^2+px+q\), seek sum \(p\) and product \(q\).
Restriction note
Write excluded denominator values before cancellation; the restriction never disappears.
4Equation and graph shortcuts
Clear fractions once
Multiply the entire equation by the denominator LCM, not selected terms.
Eliminate strategically
Choose the variable whose coefficients need the smallest multipliers.
Intercept graph
\[ax+by=c\Rightarrow\left(\frac ca,0\right),\left(0,\frac cb\right)\]
5Surd shortcuts
Extract the largest square
\[\sqrt{ka^2}=|a|\sqrt k\]
Like-surds first
Simplify every radical before deciding which terms combine.
Conjugate denominator
For a two-term radical denominator, multiply numerator and denominator by its sign-flipped partner.
SSC algebra speed dashboard
Classify the pattern, note restrictions, choose an identity or equation move, simplify, then substitute back.
\[\text{pattern}\to\text{domain}\to\text{transform}\to\text{simplify}\to\text{check}\]
04DON'T RUSH
The SSC / TCS Traps — Red Flags 🚩
Distractors lose the middle term, cancel across addition, forget domains, or treat an identity as a single-value equation.
Trap 1: middle term omitted or sign lost
- \((a\pm b)^2\) always contains the middle term \(\pm2ab\).
- The last term is \(+b^2\) in both square identities.
- A minus inside a square does not make the last square negative.
Trap 2: terms cancelled instead of factors
- Cancellation is division of common factors, not deletion of matching-looking terms.
- Factor the entire numerator and denominator first.
- Keep every original denominator restriction.
Trap 3: square root sign/domain errors
- \(\sqrt{x^2}=|x|\), not always \(x\).
- The principal root is non-negative; solving a square equation can give two signs.
- \(\sqrt{a+b}\ne\sqrt a+\sqrt b\).
Operation applied to one term only
When clearing a denominator or distributing a minus, apply the operation to every affected term.
Extraneous or lost solution
Division by a variable expression can lose zero cases; squaring can add invalid cases. Check the original equation.
Graph intercept swapped
For the x-intercept set \(y=0\); for the y-intercept set \(x=0\).
05LOCK IT IN
Memory Hooks & Mnemonics
Attach identities and equation rules to shapes, mirrors and balance scales.
“Square: first, twice-product, last”
\((a\pm b)^2=a^2\pm2ab+b^2\); only the middle sign changes.
“Difference of squares opens two doors”
One factor keeps the minus; its partner uses the plus.
“Same move, both sides”
An equation remains balanced only when the same valid operation is applied on both sides.
“Factors cancel; terms do not”
Factor complete expressions before cancelling, and preserve excluded values.
“Two equations meet at one point”
A unique simultaneous solution is the coordinate where both lines intersect.
“Conjugate changes one sign, not the design”
Keep all terms and flip only the middle sign to create a difference of squares.
Like termssame variable powers
Square identityfirst² ± twice product + last²
Factorizationreverse expansion
Equationbalance both sides
Line solutionpoint satisfying equation
Surd conjugateflip middle sign