Algebra (Linear & Quadratic Equations)

Linear equations, standard identities, and quadratic expressions beyond the standalone Quantity-Comparison format — rounding out the Quant syllabus.

📖 ~9 min read 🏦 Banking Quantitative Aptitude

Introduction

This final chapter rounds out the algebra foundation — linear equations in one or two variables, and the standard identities used to simplify algebraic expressions quickly, which appear scattered across Simplification, DI, and word-problem chapters.

Standard Identities

Identity	Expansion
(a+b)²	a² + 2ab + b²
(a−b)²	a² − 2ab + b²
a² − b²	(a+b)(a−b)
(a+b)² − (a−b)²	4ab
(a+b)² + (a−b)²	2(a² + b²)
a³+b³	(a+b)(a²−ab+b²)

Q. Solve for x and y: 2x + 3y = 12 and x − y = 1.

From the second equation: x = y+1. Substitute into the first: 2(y+1)+3y = 12 → 2y+2+3y = 12 → 5y = 10 → y = 2. Then x = 2+1 = 3. So x = 3, y = 2.

Q. If a + b = 12 and ab = 32, find a² + b².

a² + b² = (a+b)² − 2ab = 12² − 2×32 = 144 − 64 = 80.

⚠️ Key Insight: When a question gives you (a+b) and ab (or similar sums/products) instead of individual values, always check if a standard identity can get you to the answer directly — avoid solving for a and b individually unless required.

💡 Exam Tip: For two-variable linear equations, the substitution method (as in the example above) is usually faster than elimination when one equation already isolates a variable cleanly.

✅ Practice Focus: Solving simultaneous linear equations · Applying standard identities to avoid full expansion · Age/mixture-style word problems translated into linear equations.

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