1. Algebra: Identities, Linear and Quadratic Equations
SSC CGL and CHSL ask many algebra questions based on standard identities. Bank exams (IBPS and SBI PO and Clerk) ask quadratic equation comparisons, where you solve two equations and compare x and y. This lesson covers both, with step-by-step examples.
Important identities
- (a + b)² = a² + 2ab + b². (a − b)² = a² − 2ab + b². a² − b² = (a + b)(a − b).
- a³ + b³ = (a + b)(a² − ab + b²). a³ − b³ = (a − b)(a² + ab + b²). (a + b)³ = a³ + b³ + 3ab(a + b).
- If a + b + c = 0, then a³ + b³ + c³ = 3abc.
- If a + 1/a = k, then a² + 1/a² = k² − 2 and a³ + 1/a³ = k³ − 3k. Example: a + 1/a = 5 → a² + 1/a² = 25 − 2 = 23.
- If a − 1/a = k, then a² + 1/a² = k² + 2. Example: a − 1/a = 4 → 16 + 2 = 18.
- Worked example: x + y = 10 and xy = 21. Then x² + y² = (x + y)² − 2xy = 100 − 42 = 58.
Linear equations in two variables
- Solve by elimination: make the coefficient of one variable equal, then add or subtract the equations.
- Worked example: 3x + 2y = 16 and x − y = 2. Step 1: multiply the second equation by 2: 2x − 2y = 4. Step 2: add: 5x = 20, so x = 4. Step 3: y = x − 2 = 2.
- Check your answer: 3 × 4 + 2 × 2 = 16. Correct.
- Shortcut: in MCQs, put the options into both equations. Only the correct pair fits both.
Quadratic equations and bank-style comparison
- For ax² + bx + c = 0: sum of roots = −b/a, product of roots = c/a. Find two numbers whose product is c (when a = 1) and whose sum is −b.
- Worked example: x² − 7x + 12 = 0. Product 12, sum 7 → 3 and 4. Roots: x = 3, 4.
- Bank comparison: solve both equations, then compare every x root with every y root. Example: I. x² − 7x + 12 = 0 (x = 3, 4). II. y² − 3y + 2 = 0 (y = 1, 2). Every x is bigger than every y, so x > y.
- If some x values equal some y values and the rest are bigger, the answer is x ≥ y. If some x are bigger and some are smaller, the answer is 'relationship cannot be established'.
- Sign rule: if the equation is x² + bx + c with b and c both positive, both roots are negative. If b is negative and c is positive, both roots are positive.
Quick facts
- (a + b)²
- a² + 2ab + b²
- a² − b²
- (a + b)(a − b)
- a + 1/a = k
- a² + 1/a² = k² − 2; a³ + 1/a³ = k³ − 3k
- a + b + c = 0
- a³ + b³ + c³ = 3abc
- Quadratic ax² + bx + c
- Sum of roots −b/a, product c/a
- Bank comparison
- Compare every x root with every y root
💡 In SSC algebra questions, try putting simple values like a = 1 or x = 0 that satisfy the condition. Often this gives the answer without any algebra.
📝 5-question quiz on this lesson: enrol free to take it and track your score.
