Algebra

Algebra

🟢 Lite

Key Formula/Rule

Linear: ax + b = 0 → x = –b/a Quadratic: ax² + bx + c = 0 → x = [–b ± √(b²–4ac)] / 2a

Memory Trick

“Quadratic Formula Dance” 💃 — Stand on –b, spread your arms to ±, square the building (b²), subtract 4ac, then split (divide by 2a)! Left arm = +, Right arm = –.

1-Sentence Summary

Algebra finds unknowns — one variable at a time in linear equations, two solutions at once in quadratics, and patterns that grow (or shrink) steadily in progressions.

Linear Equations (ax + b = 0)

Simple form: ax + b = 0 → x = –b/a

Word to equation translation:

• “A number plus 5 equals 12” → x + 5 = 12 → x = 7
• “3 times a number is 21” → 3x = 21 → x = 7
• “A number divided by 4 is 9” → x/4 = 9 → x = 36
• “The difference between a number and 7 is 3” → x − 7 = 3 → x = 10

Simultaneous linear equations (2 variables): Method 1 — Substitution: Express x or y from one equation, substitute in the other Method 2 — Elimination: Multiply equations so one variable has same coefficient, then subtract/add Method 3 — Cross-multiplication (for ax + by = c, dx + ey = f):

x/(bf−ce) = y/(cd−af) = 1/(ae−bd)

Quadratic Equations (ax² + bx + c = 0)

Discriminant: D = b² − 4ac

• D > 0: Two distinct real roots
• D = 0: Two equal real roots (repeated)
• D < 0: No real roots (complex/imaginary)

Roots: x = (–b ± √D) / 2a

Quick factorisation trick: Look for two numbers that multiply to (a×c) and add to b. For x² + 5x + 6 = 0 → find numbers that multiply to 6 and add to 5 → 2 and 3 → (x + 2)(x + 3) = 0 → x = –2 or x = –3

Sum and product of roots (without finding them):

• Sum = –b/a
• Product = c/a

⚡ CUET Tip: If you can’t factor a quadratic quickly, use the formula — it’s always reliable. The discriminant shortcut tells you how many real roots exist without fully solving.

Arithmetic Progressions (AP)

General term: aₙ = a + (n–1)d

• a = first term, d = common difference

Sum of n terms: Sₙ = n/2 × [2a + (n–1)d] = n(a + aₙ)/2

Quick check: Difference between consecutive terms must be constant (d).

Geometric Progressions (GP)

General term: aₙ = ar⁽ⁿ⁻¹⁾

• a = first term, r = common ratio

Sum of n terms: Sₙ = a(rⁿ − 1)/(r − 1) for r ≠ 1

GP condition: Ratio between consecutive terms is constant.

⚡ CUET Exam Tip: CUET algebra questions focus on linear equations, quadratic factorisation, and AP/Gp identification. Time yourself on solving speed — target 45 seconds per linear equation and 90 seconds per quadratic. If factorisation takes more than 30 seconds, switch to the quadratic formula.

🟡 Standard

Concept

Linear Equations — One Solution, Always

A linear equation in one variable (ax + b = 0) always has exactly one solution: x = –b/a. The graph is a straight line crossing the x-axis at that point.

With TWO linear equations in two variables, you have three possibilities:

• One solution — lines intersect at one point
• No solution — lines are parallel (same slope, different intercept)
• Infinite solutions — lines are actually the same line

To solve, use either substitution (solve one for a variable, plug into the other) or elimination (multiply equations so one variable cancels when added).

Quadratic Equations — The Discriminant is Your Friend

The discriminant D = b² – 4ac tells you everything about the roots before you find them:

• D > 0: Two distinct real roots
• D = 0: Two equal real roots (repeated root)
• D < 0: No real roots (complex roots exist, but not in CUET scope)

The quadratic formula gives you the roots directly. But many CUET questions let you factor the quadratic instead — much faster when it works! If x² – 5x + 6 = 0, think: “what two numbers multiply to +6 and add to –5?” → –2 and –3. So (x–2)(x–3) = 0, giving x = 2 or 3.

Vieta’s Formulas — Sum and Product of Roots

For ax² + bx + c = 0 with roots α and β:

• α + β = –b/a (sum of roots)
• αβ = c/a (product of roots)

This is HUGE for problems that ask for expressions involving roots without finding the roots themselves.

Progressions — Patterns That Behave Predictably

An Arithmetic Progression (AP) has a constant difference d between consecutive terms:

• aₙ = a + (n–1)d
• Sum of n terms: Sₙ = n/2[2a + (n–1)d] = n(a + l)/2 where l is the last term

A Geometric Progression (GP) has a constant ratio r:

• aₙ = ar^(n–1)
• Sum of n terms: Sₙ = a(r^n – 1)/(r – 1) when r ≠ 1

Key Formulas

Formula	Use
x = –b/a	Solving linear equation ax + b = 0
x = [–b ± √(b²–4ac)]/2a	Quadratic formula
D = b² – 4ac	Discriminant
α + β = –b/a	Sum of quadratic roots
αβ = c/a	Product of quadratic roots
aₙ = a + (n–1)d	nth term of AP
Sₙ = n/2[2a + (n–1)d]	Sum of n terms of AP
aₙ = ar^(n–1)	nth term of GP
Sₙ = a(r^n – 1)/(r – 1)	Sum of n terms of GP (r ≠ 1)

Worked Example

Q: If α and β are roots of x² – 7x + 10 = 0, find α³ + β³.

Step 1: From Vieta: α + β = 7, αβ = 10 Step 2: Use identity: α³ + β³ = (α + β)³ – 3αβ(α + β) Step 3: = 7³ – 3(10)(7) = 343 – 210 = 133

Answer: 133

Common Errors

• Forgetting to divide by a in quadratic formula → It’s 2a in the denominator, not just 2!
• Sign errors in Vieta’s formulas → α + β = –b/a, αβ = c/a — watch the negative sign for sum
• Using wrong GP sum formula for r < 1 → For r < 1, Sₙ = a(1 – r^n)/(1 – r) is equivalent — same thing
• Confusing AP and GP → AP: add d each time; GP: multiply by r each time

🔴 Extended

Full Concept

Linear Equations — Elimination vs Substitution

For two equations in two variables:

• Elimination works best when coefficients match or can be made to match easily
• Substitution works best when one equation gives a variable easily (e.g., y = something in x)

Example: 2x + 3y = 8 x – 2y = –3

Elimination approach: Multiply second equation by 2: 2x – 4y = –6. Subtract from first: (2x+3y) – (2x–4y) = 8–(–6) → 7y = 14 → y = 2. Then x = –3 + 2(2) = 1.

The Discriminant — Your Pre-Flight Check

Before solving a quadratic with the formula, always check D:

• D < 0? → No real roots. Save yourself the algebra!
• D is a perfect square? → Roots are rational → factoring will work
• D is not a perfect square? → Roots are irrational → use the formula

Why Vieta’s Formulas Work

For ax² + bx + c = 0 with roots α and β:

From the quadratic formula: α = [–b + √D]/2a, β = [–b – √D]/2a

Adding: α + β = [–b + √D – b – √D]/2a = –2b/2a = –b/a ✓ Multiplying: αβ = [(–b)² – (√D)²]/(2a)² = [b² – D]/4a²

But D = b² – 4ac, so αβ = [b² – (b² – 4ac)]/4a² = 4ac/4a² = c/a ✓

Arithmetic Progression — Beyond the Basics

The sum formula Sₙ = n/2[2a + (n–1)d] can also be written as n(a + l)/2 where l is the last term. This second form is often easier when you know the first and last terms.

If you need the sum of first n natural numbers: that’s AP with a=1, d=1 → S = n(n+1)/2.

Three consecutive terms in AP: If a–d, a, a+d are three consecutive terms (they center around a with common difference d). This form is super useful for word problems!

Geometric Progression — Growth and Decay

GP is where things get interesting:

• r > 1: Terms grow exponentially (population growth, compound interest)
• 0 < r < 1: Terms shrink toward zero (radioactive decay, depreciation)
• r < 0: Terms alternate sign (oscillating sequences)

The GP sum formula Sₙ = a(r^n – 1)/(r – 1) works for r ≠ 1. When r > 1, this is fine. When 0 < r < 1, r^n becomes very small, so Sₙ ≈ a/(1 – r) as n → ∞ — this is the sum to infinity.

When r < 0, the alternating sign causes oscillation — there’s no sum to infinity unless |r| < 1, in which case S∞ = a/(1 – r) still holds.

Harmonic Progression — The Neglected Sibling

HP is defined as the reciprocals of an AP. So 1/a, 1/(a+d), 1/(a+2d), … form an HP.

To solve HP problems: convert to AP by taking reciprocals, solve the AP problem, then convert back.

Multiple Approaches

For Quadratic Word Problems:

1. Translate to equation — identify what x represents
2. Set up quadratic — form the equation
3. Solve — factor or use formula
4. Check — does the answer make sense in context?

For “find sum of roots expressions”:

1. Try to express the desired quantity in terms of α + β and αβ
2. Use Vieta’s formulas directly
3. Avoid finding individual roots unless absolutely necessary

CUET-Level Problems

Q1: The difference between roots of x² – 2kx + k² + k – 5 = 0 is 4. Find k.

Working:

• Let roots be α and β: α + β = 2k, αβ = k² + k – 5
• Difference: |α – β| = √[(α+β)² – 4αβ] = √[4k² – 4(k² + k – 5)] = √[4k² – 4k² – 4k + 20] = √(20 – 4k)
• Given: √(20 – 4k) = 4 → 20 – 4k = 16 → 4k = 4 → k = 1

Answer: k = 1

Q2: Insert 3 numbers between 3 and 48 so that the sequence becomes a GP.

Working:

• Let the GP be: 3, G₁, G₂, G₃, 48
• This is a 5-term GP with a = 3, a₅ = 48, n = 5
• a₅ = ar⁴ → 48 = 3 × r⁴ → r⁴ = 16 → r = 2 (or r = –2)
• For r = 2: terms are 3, 6, 12, 24, 48
• For r = –2: terms are 3, –6, 12, –24, 48

Answer: (3, 6, 12, 24, 48) or (3, –6, 12, –24, 48)

Tricky Cases

• Two linear equations with no solution → Coefficients are proportional but constants aren’t: a₁/a₂ = b₁/b₂ ≠ c₁/c₂
• Two linear equations with infinite solutions → All coefficients proportional: a₁/a₂ = b₁/b₂ = c₁/c₂
• Using GP sum to infinity when r ≥ 1 → Infinity sum ONLY exists when |r| < 1
• HP problems → Always convert to AP first, solve, then convert back
• Roots not real in quadratic → If D < 0 and question asks for real roots → answer is “no real roots”

Content adapted based on your selected roadmap duration.