Algebra — Study Notes for SSC MTS

Overview

Algebra in SSC MTS Paper 1 focuses on solving linear equations (one or two variables), basic quadratic equations, and applying algebraic identities to simplify expressions. This topic typically accounts for 2–3 questions worth 2–3 marks. Unlike advanced algebra, MTS tests your ability to manipulate variables, factorize simple expressions, and substitute values correctly. Mastery here requires recognizing patterns quickly (which identity applies?), avoiding calculation errors (especially sign mistakes), and solving equations methodically. Questions often interweave with arithmetic—for example, translating a word problem into an equation or verifying an identity numerically. Solid algebra skills also accelerate work in topics like Time-Work, Profit-Loss, and Mixture problems where you set up equations to find unknowns. The syllabus stops short of complex polynomials or simultaneous equations beyond two variables, so your preparation should emphasize speed and accuracy on standard forms.

Key Concepts

• **Linear equation in one variable**: An equation of the form ax + b = c, where a ≠ 0. Solve by isolating x through inverse operations (add/subtract constants, then divide by coefficient).
• **Linear equation in two variables**: Equations like 2x + 3y = 12. MTS questions usually give two such equations (simultaneous equations) and ask for x and y. Solve by substitution (express one variable in terms of the other) or elimination (add/subtract equations to cancel one variable).
• **Quadratic equation**: An equation of the form ax² + bx + c = 0, where a ≠ 0. For MTS, factorization is the primary method—split the middle term or recognize perfect squares. The discriminant (b² − 4ac) and quadratic formula appear rarely but are worth knowing for non-factorable cases.
• **Algebraic identities**: Pre-memorized expansion formulas that simplify expressions without multiplying term-by-term. The exam tests both direct substitution (given a + b = 5 and ab = 6, find a² + b²) and reverse recognition (factor x² − y² immediately as (x + y)(x − y)).
• **Substitution and simplification**: Replace variables with given numerical values and compute the result. Watch for negative signs and fractions—errors here cost marks despite correct method.
• **Factorization**: Writing an expression as a product of simpler expressions. Common techniques include taking out the greatest common factor, grouping terms, and applying identities.
• **Transposition**: Moving terms across the equals sign by changing their sign. This is the workhorse of equation solving—many mistakes happen when students forget to flip the sign.

Formulas / Key Facts

**Standard Algebraic Identities** (must memorize):

1. **(a + b)² = a² + 2ab + b²** 2. **(a − b)² = a² − 2ab + b²** 3. **a² − b² = (a + b)(a − b)** (difference of squares) 4. **(a + b)³ = a³ + 3a²b + 3ab² + b³** or **a³ + b³ + 3ab(a + b)** 5. **(a − b)³ = a³ − 3a²b + 3ab² − b³** or **a³ − b³ − 3ab(a − b)** 6. **a³ + b³ = (a + b)(a² − ab + b²)** 7. **a³ − b³ = (a − b)(a² + ab + b²)** 8. **(x + a)(x + b) = x² + (a + b)x + ab**

**Derived useful relations**:

• **a² + b² = (a + b)² − 2ab** = **(a − b)² + 2ab**
• **a⁴ + a²b² + b⁴ = (a² + ab + b²)(a² − ab + b²)**

**Quadratic formula** (if factorization fails): For ax² + bx + c = 0, roots are **x = [−b ± √(b² − 4ac)] / 2a**

Worked Examples

**Example 1**: Solve for x: **5x − 3 = 2x + 9**

• Step 1: Transpose 2x to the left: 5x − 2x − 3 = 9 → 3x − 3 = 9
• Step 2: Transpose −3 to the right: 3x = 9 + 3 = 12
• Step 3: Divide by 3: x = 4

**Example 2**: Solve the system **x + y = 10** and **2x − y = 5**

• Method (Elimination): Add both equations to cancel y:

(x + y) + (2x − y) = 10 + 5 → 3x = 15 → x = 5

• Substitute x = 5 into first equation: 5 + y = 10 → y = 5
• Solution: x = 5, y = 5

**Example 3**: Factorize **x² + 5x + 6**

• Need two numbers that multiply to 6 and add to 5: those are 2 and 3.
• Write as **(x + 2)(x + 3)**
• Verify: (x + 2)(x + 3) = x² + 3x + 2x + 6 = x² + 5x + 6 ✓

**Example 4**: Given **a + b = 7** and **ab = 10**, find **a² + b²**

• Use identity: a² + b² = (a + b)² − 2ab
• Substitute: a² + b² = 7² − 2(10) = 49 − 20 = **29**

**Example 5**: Simplify **(x + 3)² − (x − 3)²**

• Recognize difference of squares: A² − B² = (A + B)(A − B), where A = x + 3, B = x − 3
• (A + B) = (x + 3) + (x − 3) = 2x
• (A − B) = (x + 3) − (x − 3) = 6
• Product = 2x × 6 = **12x**

**Example 6**: Solve **x² − 7x + 12 = 0**

• Factorize: find two numbers that multiply to 12 and add to −7: −3 and −4
• (x − 3)(x − 4) = 0
• Solutions: x = 3 or x = 4

Common Mistakes

• **Sign errors during transposition**: Students write 5x − 3 = 12, then 5x = 12 − 3 instead of 12 + 3. Fix: When moving a term across =, always reverse its sign (− becomes +, + becomes −).
• **Confusing (a − b)² with a² − b²**: Writing (a − b)² = a² − b² misses the middle term −2ab. Fix: Memorize the full identity (a − b)² = a² − 2ab + b², and note a² − b² is a separate identity (difference of squares).
• **Incorrect middle-term splitting**: Attempting to factor x² + 7x + 10 by choosing numbers that don't multiply to 10 or don't add to 7 (e.g., 3 and 4). Fix: Always verify both conditions—product equals constant term, sum equals coefficient of x.
• **Forgetting to check both roots**: Solving x² = 9 and writing x = 3 only, missing x = −3. Fix: Remember equations of degree n have n roots; quadratics yield two solutions (which may coincide).
• **Substitution errors with negative values**: Given x = −2, computing x² − 3x as (−2)² − 3(−2) but writing 4 − 6 = −2 instead of 4 + 6 = 10. Fix: Write negatives explicitly in parentheses and perform operations carefully.

Quick Reference

• **Linear equation one variable**: Isolate x by inverse operations; always transpose with sign change.
• **Simultaneous equations**: Use elimination (add/subtract equations) or substitution (solve one, plug into other).
• **(a + b)² = a² + 2ab + b²** and **(a − b)² = a² − 2ab + b²**—know both cold, they're 40% of identity questions.
• **a² − b² = (a + b)(a − b)**—instant factorization for difference of squares.
• **Factorize quadratics**: Find pair of numbers (product = c, sum = b in x² + bx + c).
• **a² + b² = (a + b)² − 2ab**—key shortcut when sum and product are given.