Algebra

🟢 Lite — Quick Review (1h–1d)

Rapid summary for last-minute revision before your exam.

Algebra — Quick Facts for SSC CGL

Basic Identities:

• (a + b)² = a² + 2ab + b²
• (a − b)² = a² − 2ab + b²
• (a + b)(a − b) = a² − b²
• (a + b)³ = a³ + 3a²b + 3ab² + b³ = a³ + b³ + 3ab(a + b)
• (a − b)³ = a³ − 3a²b + 3ab² − b³
• a³ + b³ = (a + b)(a² − ab + b²)
• a³ − b³ = (a − b)(a² + ab + b²)
• (a + b + c)² = a² + b² + c² + 2(ab + bc + ca)

Quadratic Equation: ax² + bx + c = 0 (where a ≠ 0)

Quadratic Formula (Shridharacharya’s Formula): $$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$$

Discriminant (D = b² − 4ac):

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

Sum and Product of Roots:

• Sum of roots (α + β) = −b/a
• Product of roots (αβ) = c/a

Nature of Roots Based on Discriminant: For real and equal roots: D = 0 For real roots: D ≥ 0 For roots to be reciprocal: αβ = 1 → c/a = 1 → c = a For roots to be opposite signs: αβ < 0 → c/a < 0 → c and a have opposite signs

⚡ Exam tip: When solving quadratic equations by factorisation, write the equation in the form ax² + bx + c = 0, then find two numbers that multiply to ac and add to b. For x² + 5x + 6 = 0: numbers are 2 and 3 (2+3=5, 2×3=6) → (x+2)(x+3)=0 → x = −2 or −3.

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🟡 Standard — Regular Study (2d–2mo)

Standard content for students with a few days to months.

Algebra — SSC CGL Study Guide

Types of Quadratic Equations:

Pure Quadratic: bx² + c = 0 (no x term) Example: x² − 9 = 0 → x² = 9 → x = ±3

Complete Quadratic: ax² + bx + c = 0

Equation reducible to quadratic:

1. Reciprocal equation: If 1/x replaces x, the equation becomes quadratic Example: 1/(x+1) + 1/(x+2) = 2/(x+3)
2. Exponential form: When a^x appears Example: 3^(2x) − 4 × 3^x + 3 = 0 → substitute 3^x = y → y² − 4y + 3 = 0
3. Square-root equations: √(ax + b) = cx + d → Square both sides, check extraneous solutions

Progressions:

Arithmetic Progression (AP):

• a, a+d, a+2d, …, a+(n−1)d
• nth term: aₙ = a + (n−1)d
• Sum of n terms: Sₙ = n/2 × (2a + (n−1)d) = n/2 × (a + l) where l = last term

Geometric Progression (GP):

• a, ar, ar², …, ar^(n−1)
• nth term: aₙ = ar^(n−1)
• Sum of n terms: Sₙ = a(r^n − 1)/(r − 1) if r > 1; or a(1 − r^n)/(1 − r) if r < 1
• Sum to infinity: S∞ = a/(1 − r), only if |r| < 1

Harmonic Progression (HP):

• Reciprocals of HP terms form an AP
• nth term of HP: 1/aₙ = 1/a + (n−1)d where d = (1/b − 1/a) if a and b are first two terms

Example: Find the 10th term of the AP: 3, 7, 11, …

• a = 3, d = 4, n = 10
• a₁₀ = 3 + (10−1) × 4 = 3 + 36 = 39

Example: Sum of first 20 odd numbers:

• Sequence: 1, 3, 5, … (odd numbers)
• n = 20, a = 1, d = 2
• S₂₀ = 20/2 × (2×1 + (20−1)×2) = 10 × (2 + 38) = 10 × 40 = 400
• Alternative: Sum of first n odd numbers = n² = 20² = 400 ✓

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🔴 Extended — Deep Study (3mo+)

Comprehensive coverage for students on a longer study timeline.

Algebra — Comprehensive Notes

Advanced Algebra Techniques:

Simultaneous Equations: Two equations in two variables:

• Substitution method: Express one variable in terms of the other, substitute
• Elimination method: Multiply equations to make coefficients of one variable equal, then add/subtract
• Cross-multiplication (for ax + by = c and dx + ey = f): x/(bf − ce) = y/(cd − af) = 1/(ae − bd)

Word Problems — Worked Examples:

Example 1 (Numbers): The sum of a two-digit number and its reverse is 154. The tens digit is 4 less than the units digit. Find the number.

• Let tens digit = x, units digit = y
• Number = 10x + y; Reverse = 10y + x
• (10x+y) + (10y+x) = 154 → 11x + 11y = 154 → x + y = 14
• y − x = 4 (tens digit is 4 less than units digit → y > x)
• Solving: x + y = 14, y − x = 4 → 2y = 18 → y = 9, x = 5
• Number = 59

Example 2 (Time and Work): A can do a work in 15 days, B in 20 days. A works for 5 days and then leaves. How long does B take to finish the remaining work?

• A’s 1 day work = 1/15
• B’s 1 day work = 1/20
• A works 5 days → completes 5 × 1/15 = 1/3 of work
• Remaining work = 2/3
• B takes (2/3) / (1/20) = 40/3 = 13⅓ days

Example 3 (Boats and Streams): A boat goes 20 km downstream in 2 hours and the same distance upstream in 5 hours. Find the speed of the boat in still water and the speed of the stream.

• Downstream speed = 20/2 = 10 km/h
• Upstream speed = 20/5 = 4 km/h
• Let boat speed = b, stream speed = s
• b + s = 10; b − s = 4
• Adding: 2b = 14 → b = 7 km/h; s = 3 km/h

Example 4 (Percentage): A shopkeeper sells an article at 20% profit. If he had bought it at 10% less and sold it at ₹10 less, he would have made a 30% profit. Find the cost price.

• Let original CP = ₹100
• Original SP = ₹120
• New CP = ₹90; Desired SP for 30% profit = 90 × 1.3 = ₹117
• Given new SP = 120 − 10 = ₹110
• But ₹110 ≠ ₹117… Let CP = x
• Original SP = 1.2x
• New CP = 0.9x; New SP = 1.2x − 10
• Profit = (1.2x − 10 − 0.9x)/0.9x × 100 = 30
• (0.3x − 10)/0.9x × 100 = 30 → (0.3x − 10) = 0.27x → 0.03x = 10 → x = 1000/3 ≈ 333.33
• CP = ₹333.33; Original SP = ₹400; New SP = ₹390; New CP = ₹300; Profit = 90/300 = 30% ✓

Quadratic Equation — Formation from Roots: If roots are α and β, the equation is: x² − (α + β)x + αβ = 0 Example: Roots are 3 and −2

• Sum = 1, Product = −6
• Equation: x² − x − 6 = 0

Maxima and Minima of Quadratic: For y = ax² + bx + c:

• Vertex occurs at x = −b/(2a)
• If a > 0: Minimum value = (4ac − b²)/(4a)
• If a < 0: Maximum value = (4ac − b²)/(4a)

Logarithms (Useful for Solving Exponential Equations):

• If a^x = N → logₐN = x
• log(ab) = log a + log b
• log(a/b) = log a − log b
• log(a^n) = n log a
• log₁₀a + log₁₀b = log₁₀(ab)
• Change of base: logₐN = log_bN / log_ba

NEET/SSC Pattern Analysis: Algebra is the backbone of SSC CGL Tier-II Quantitative Aptitude. High-frequency areas: quadratic equations (roots, discriminant, sum/product), word problems leading to quadratic, AP/GP sum and nth term, simultaneous equations, and logarithm basics. Questions on “ages” and “numbers” are particularly common.

⚡ SSC CGL 2022 Qn: If α and β are the roots of x² − 5x + 6 = 0, then find the value of α² + β². Answer: α + β = 5, αβ = 6. α² + β² = (α + β)² − 2αβ = 25 − 12 = 13.