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Algebra — The Language of Unknowns

Learning Goals

By the end of this chapter, you will be able to:

• Master identities and factorisation without memorising blindly
• Solve linear, quadratic and simultaneous equations with speed
• Handle inequalities, modulus and surds confidently
• Tackle polynomials, remainder/factor theorems and progressions (AP/GP)
• Spot the traps examiners set in SSC CGL Tier-II, IBPS PO Mains, CAT and CSAT

The Hook: The Number That Terrified a Civilisation

Around 500 BCE, a Greek mathematician named Hippasus made a discovery so disturbing that, legend says, his own school — the Pythagoreans — drowned him at sea for it. His crime? He proved that some numbers cannot be written as a fraction. The square root of 2 was "irrational", and the world of neat, tidy numbers cracked open forever.

Why start here? Because Algebra is exactly this: the moment mathematics stops being about fixed numbers and starts being about the unknown. The moment you write *x*, you are doing what terrified the ancients — admitting you don't know something, and then *systematically hunting it down*.

Every competitive exam you will face — SSC CGL, IBPS PO, CAT, CSAT — is secretly testing one skill: can you convert a messy word problem into a clean equation, and solve it fast? This chapter gives you that weapon.

"But I Hate Algebra…"

Let's be honest. Most aspirants carry a school-time scar: pages full of *x* and *y*, a teacher saying "just transpose it", and no idea *why*. Here's the truth nobody told you:

Algebra is not about letters. It is about balance.

An equation is a weighing scale. Whatever you do to one side, you must do to the other. That's it. Every rule — transposition, cross-multiplication, elimination — is just a shortcut for keeping the scale balanced. Once you see this, the fear dissolves.

🧠 Active Recall: Before reading further, close your eyes and answer — why are we allowed to "move" a term from LHS to RHS by changing its sign? (Hint: what are we *actually* doing to both sides?)

Chunk 1: The Golden Identities

Everything in advanced algebra stands on a handful of identities. Don't memorise them as symbols — memorise them as *patterns*.

• (a + b)² = a² + 2ab + b²
• (a − b)² = a² − 2ab + b²
• a² − b² = (a + b)(a − b)
• (a + b)³ = a³ + b³ + 3ab(a + b)
• (a − b)³ = a³ − b³ − 3ab(a − b)
• a³ + b³ = (a + b)(a² − ab + b²)
• a³ − b³ = (a − b)(a² + ab + b²)
• a³ + b³ + c³ − 3abc = (a + b + c)(a² + b² + c² − ab − bc − ca)

The exam favourite: if a + b + c = 0, then a³ + b³ + c³ = 3abc. This single result cracks many "impossible-looking" questions in SSC and CAT.

The workhorse trick: from (a + b)² and (a − b)², we get
(a + b)² − (a − b)² = 4ab and (a + b)² + (a − b)² = 2(a² + b²).

So if x + 1/x = 5, then x² + 1/x² = 5² − 2 = 23, and x³ + 1/x³ = 5³ − 3×5 = 110. Notice the pattern: square → subtract 2; cube → subtract 3 times the base. This is one of the most repeated algebra patterns in SSC exams.

🧠 Active Recall: If x − 1/x = 4, what is x² + 1/x²? (Careful — the sign flips the adjustment. Answer: 4² + 2 = 18.)

Chunk 2: Linear Equations — One Unknown, One Truth

A linear equation in one variable, ax + b = 0, has exactly one solution: x = −b/a. Simple. But exams rarely give you the equation — they give you a *story*: ages, work, money, mixtures.

The 3-step translation method:

1. Name the unknown (let the present age be x).
2. Translate each sentence into an equation piece by piece.
3. Solve and verify against the story, not just the algebra.

For two variables, you need two independent equations. Solve by:

• Substitution (express one variable, plug in)
• Elimination (match coefficients, add/subtract)
• Cross-multiplication for a₁x + b₁y + c₁ = 0 and a₂x + b₂y + c₂ = 0

Consistency check (a CSAT/CGL favourite):

• a₁/a₂ ≠ b₁/b₂ → unique solution
• a₁/a₂ = b₁/b₂ = c₁/c₂ → infinitely many solutions
• a₁/a₂ = b₁/b₂ ≠ c₁/c₂ → no solution

🧠 Active Recall: Two equations represent two lines. What does "no solution" mean geometrically? (Parallel lines — they never meet.)

Chunk 3: Quadratic Equations — Where the Real Game Begins

ax² + bx + c = 0 (a ≠ 0). Three tools:

1. Factorisation — fastest when roots are integers.
2. Quadratic formula: x = [−b ± √(b² − 4ac)] / 2a
3. Discriminant analysis: D = b² − 4ac

• D > 0 → two distinct real roots
• D = 0 → equal roots
• D < 0 → no real roots
• D a perfect square (with rational coefficients) → rational roots

Vieta's relations (memorise like your name):

• Sum of roots = −b/a
• Product of roots = c/a

These two lines solve many quadratic questions without ever finding the roots. "The sum of roots is 7 and product is 12" → roots are 3 and 4, equation is x² − 7x + 12 = 0. Done in seconds.

Sign tricks: If both roots are positive → sum positive, product positive. If roots have opposite signs → product negative (c/a < 0). Examiners love asking about the *nature* of roots precisely because students ignore these shortcuts.

🧠 Active Recall: Without solving, what is the sum of roots of 2x² − 9x + 4 = 0? (Answer: 9/2.) And what does D = 81 − 32 = 49 tell you? (Perfect square → rational roots.)

Chunk 4: Polynomials, Remainder & Factor Theorems

Remainder Theorem: When polynomial p(x) is divided by (x − a), the remainder is simply p(a). No long division needed — just substitute.

Factor Theorem: (x − a) is a factor of p(x) if and only if p(a) = 0.

Example: Is (x − 2) a factor of x³ − 3x² + 4x − 4? Compute p(2) = 8 − 12 + 8 − 4 = 0. Yes, it is. Ten seconds, zero division.

These two theorems convert terrifying cubic problems into one-line substitutions — a classic IBPS PO Mains and CAT pattern.

🧠 Active Recall: What is the remainder when x²⁰²⁴ + 1 is divided by (x − 1)? (Just p(1) = 2.)

Chunk 5: Inequalities and Modulus

Inequalities follow equation rules with one deadly exception: multiplying or dividing by a negative number flips the sign. −2x > 6 → x < −3 (not x > −3). This single flip is the examiner's favourite trap.

Modulus basics:

• |x| < a → −a < x < a (a sandwich)
• |x| > a → x < −a or x > a (two separate wings)

Mnemonic to remember forever — "LESS is a Sandwich, MORE is a War (two sides)":
Less-than squeezes x between two values; greater-than splits the answer into two opposite regions.

For quadratic inequalities like x² − 5x + 6 < 0, factorise: (x − 2)(x − 3) < 0. The product is negative *between* the roots → 2 < x < 3. Rule: "< 0 lives between the roots, > 0 lives outside them" (when the coefficient of x² is positive).

🧠 Active Recall: Solve |2x − 3| ≤ 5. (Sandwich: −5 ≤ 2x − 3 ≤ 5 → −1 ≤ x ≤ 4.)

Chunk 6: Surds, Indices and Logarithms

Laws of indices (the non-negotiables):

• aᵐ × aⁿ = aᵐ⁺ⁿ ; aᵐ ÷ aⁿ = aᵐ⁻ⁿ ; (aᵐ)ⁿ = aᵐⁿ
• a⁰ = 1 ; a⁻ⁿ = 1/aⁿ ; a^(m/n) = ⁿ√(aᵐ)

Logarithms are just indices wearing a mask: logₐb = c means a^c = b.

• log(MN) = log M + log N
• log(M/N) = log M − log N
• log Mⁿ = n log M
• logₐb = log b / log a (change of base)

Mnemonic — "Logs turn shopping into addition": multiplication becomes addition, division becomes subtraction, powers become multiples. Whenever you see a scary product of powers, take logs and watch it melt.

🧠 Active Recall: If log₁₀2 ≈ 0.301, what is log₁₀8? (8 = 2³ → 3 × 0.301 = 0.903.)

Chunk 7: Progressions — AP and GP

Arithmetic Progression (AP): constant difference d.

• nth term: aₙ = a + (n − 1)d
• Sum of n terms: Sₙ = n/2 [2a + (n − 1)d] = n/2 (first + last)

Geometric Progression (GP): constant ratio r.

• nth term: aₙ = arⁿ⁻¹
• Sum: Sₙ = a(rⁿ − 1)/(r − 1), r ≠ 1
• Infinite GP (|r| < 1): S∞ = a/(1 − r)

Handy facts: If a, b, c are in AP → 2b = a + c. If in GP → b² = ac. The arithmetic mean of two positive numbers is always ≥ their geometric mean (AM ≥ GM) — a result that powers an entire family of maxima-minima questions in CAT.

🧠 Active Recall: The 10th term of an AP with a = 3, d = 4? (3 + 9×4 = 39.) And the sum of the infinite GP 1 + 1/2 + 1/4 + …? (1/(1 − 1/2) = 2.)

The Examiner's Trap Box

1. Sign flip in inequalities — dividing by a negative and forgetting to flip.
2. x + 1/x confusion — squaring x − 1/x and subtracting 2 instead of adding.
3. D < 0 misread — "no real roots" does not mean "no answer"; it may be exactly what the question asks.
4. |x| = 5 has two answers — x = ±5. Never drop the negative case.
5. a⁰ trap — any non-zero number to the power 0 is 1, but 0⁰ is undefined; exams exploit the edge.
6. AM ≥ GM equality — holds only when all numbers are equal; questions hide maxima right there.

Summary Table

Final Word

Algebra rewards the calm, not the clever. Every "hard" question in your exam is one of the seven chunks above wearing a costume. Your job in the exam hall is not to be brilliant — it is to recognise the costume in the first 15 seconds.

🔁 Review this chapter after 2 days. Re-attempt every Active Recall prompt without looking. Then revisit after 7 days. Spaced repetition is what converts this chapter from "read once" into "scored forever".