Algebra for Aptitude — Linear Equations, Quadratics and Identities

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

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

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.