# Algebra for Aptitude — Linear Equations, Quadratics and Identities

Source: https://skills101.in/learn/aptitude/algebra-for-aptitude — Skills101 free tutorial (Quantitative Aptitude). Updated 2026-11-09.

Algebra is a tool for turning a sentence into an equation. Many aptitude questions (ages, numbers, money) become easy once you write the equation.

## 1. An equation is a balance

![A balance with 2x + 3 on one side and 11 on the other: subtract 3 from both sides, then divide both sides by 2, to get x = 4](https://skills101.in/learn-img/aptitude/balance-equation.svg)

Whatever you do to one side, do to the other side. The goal is to get x alone.

### Example 1
*Solve 5x − 7 = 3x + 9.*

Move x terms to one side: 5x − 3x = 9 + 7 → 2x = 16 → x = **8**.

## 2. Word problems → equations

| Words | Algebra |
|---|---|
| "a number" | x |
| "5 more than a number" | x + 5 |
| "twice a number" | 2x |
| "a number decreased by 4" | x − 4 |
| "the sum of two consecutive numbers" | x + (x + 1) |
| "a two-digit number with digits a and b" | 10a + b |

### Example 2
*The sum of three consecutive odd numbers is 75. Find the largest.*

x + (x + 2) + (x + 4) = 75 → 3x + 6 = 75 → x = 23 → numbers 23, 25, 27 → largest **27**.

### Example 3 — digits
*A two-digit number is 4 times the sum of its digits. When 27 is added, the digits swap. Find the number.*

Number = 10a + b.
- 10a + b = 4(a + b) → 6a = 3b → b = 2a
- 10a + b + 27 = 10b + a → 9a + 27 = 9b → b = a + 3
- 2a = a + 3 → a = 3, b = 6 → number = **36**. (Check: 3 + 6 = 9, 4 × 9 = 36 ✓; 36 + 27 = 63 ✓)

## 3. Two equations, two unknowns

### Example 4
*5 pens and 3 notebooks cost ₹120. 2 pens and 3 notebooks cost ₹84. Find the price of a pen and a notebook.*

Subtract the second from the first: 3 pens = 36 → pen = **₹12**. Then 2 × 12 + 3n = 84 → 3n = 60 → notebook = **₹20**.

### Example 5
*x + y = 20 and x − y = 4. Find x and y.*

Add: 2x = 24 → x = **12**, y = **8**.

## 4. Quadratic equations

ax² + bx + c = 0. The fastest method in exams is **factorisation**: find two numbers that **multiply to ac** and **add to b**.

### Example 6
*Solve x² − 5x + 6 = 0.*

Two numbers that multiply to 6 and add to −5: −2 and −3 → (x − 2)(x − 3) = 0 → x = **2 or 3**.

If factors are hard, use the formula: **x = [−b ± √(b² − 4ac)] ÷ 2a**.

Useful facts for roots α and β:
- **α + β = −b/a**
- **α × β = c/a**

### Example 7
*The product of two consecutive positive numbers is 132. Find them.*

x(x + 1) = 132 → x² + x − 132 = 0 → (x + 12)(x − 11) = 0 → x = 11 → **11 and 12**.

## 5. Identities you must know

- (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)
- a³ + b³ = (a + b)(a² − ab + b²)

### Example 8
*If a + b = 10 and ab = 21, find a² + b².*

a² + b² = (a + b)² − 2ab = 100 − 42 = **58**.

### Example 9
*If x + 1/x = 5, find x² + 1/x².*

Square: x² + 2 + 1/x² = 25 → x² + 1/x² = **23**.

## Common mistakes

- Changing the sign wrongly when moving a term across "=".
- Forgetting the negative root in quadratics (and then picking the wrong one for "positive number" questions).
- Not checking the answer in the original words.

## Quick recap

- Keep the balance: same operation on both sides.
- Translate words into algebra with the table above.
- Two equations: add or subtract to remove one unknown.
- Quadratics: factorise (multiply to ac, add to b).
- Identities turn "find a² + b²" questions into one line.


Practice quiz and free certificate: https://skills101.in/learn/aptitude/algebra-for-aptitude