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Quadratic Equations & Series

Why This Chapter Can Make or Break Your Score

In almost every major exam — SSC CGL Tier-II, IBPS PO Mains, CAT, State PCS — quadratic equations and series questions appear as "quick kills." Candidates who know the patterns solve a quadratic comparison question in well under a minute; others burn 3 minutes and still pick the wrong option. The difference is not intelligence — it is knowing a handful of patterns cold. This chapter gives you exactly those patterns.

Learning goals:

1. Solve any quadratic equation by factorisation, formula, or inspection.
2. Master root–coefficient relations (sum and product of roots) and use them backwards.
3. Crack "compare x and y" questions without fully solving.
4. Identify and solve Arithmetic, Geometric, and special series (including missing/wrong number series).

Part 1: The Quadratic Equation — One Shape, Many Faces

A quadratic equation is any equation that can be written as:

ax² + bx + c = 0, where a ≠ 0.

That's it. The "quad" simply means the highest power of x is 2. Everything in this chapter flows from this one line.

There are three ways to find its solutions (called roots):

1. Factorisation — split the middle term so that ax² + bx + c breaks into two brackets.
Example: x² − 5x + 6 = 0 → (x − 2)(x − 3) = 0 → x = 2 or x = 3.

2. The quadratic formula — works always, even when factorisation fails:

x = [−b ± √(b² − 4ac)] / 2a

3. Inspection (the exam trick) — if the roots are integers, their sum is −b/a and product is c/a. Guess two numbers that fit, and you're done in seconds.

🧠 Active Recall: Without looking back — what are the three methods to solve a quadratic? What is the formula for the sum of roots?

Part 2: The Discriminant — The Equation's Fortune Teller

The expression inside the square root, D = b² − 4ac, is called the discriminant. Before you solve anything, D tells you what kind of roots to expect:

• D > 0 → two distinct real roots
• D = 0 → two equal real roots (x = −b/2a)
• D < 0 → no real roots (complex roots)

Examiners love asking "for what value of k does the equation have equal roots?" — the answer is always: set D = 0 and solve for k.

🧠 Active Recall: If D = 0, what is the single root equal to? What does D < 0 mean?

Part 3: Sum and Product of Roots — The Real Exam Weapon

For ax² + bx + c = 0 with roots α and β:

• Sum of roots: α + β = −b/a
• Product of roots: αβ = c/a

This pair of facts is the single most tested idea in this chapter. It works backwards too: if you know the sum and product, you can *build* the equation:

x² − (sum of roots)x + (product of roots) = 0

Memory trick — "S-P-S": Sum gives the Sign-flipped middle term; Product gives the last term. Say it as: *"Flip the Sum, keep the Product."*

Example: Roots are 4 and −3. Sum = 1, Product = −12. Equation: x² − (1)x + (−12) = 0 → x² − x − 12 = 0. Ten seconds, done.

🧠 Active Recall: Build the quadratic whose roots are 5 and −2. (Answer: x² − 3x − 10 = 0.)

Part 4: The "Compare x and y" Question Type

This is the signature question of bank and SSC exams:

> I. 2x² − 7x + 6 = 0
> II. 2y² − 9y + 10 = 0
> Compare x and y.

The trap: solving both equations fully. You often don't need to. Use the sign trick, which follows directly from the sum and product of roots (assuming a > 0):

• Form x² − bx + c with b > 0, c > 0: sum of roots = b > 0 and product = c > 0 → both roots are positive.
• Form x² + bx + c with b > 0, c > 0: sum of roots = −b < 0 and product = c > 0 → both roots are negative.
• c < 0 (product negative) → roots have opposite signs — one positive, one negative.

So if every root of x is positive and every root of y is negative, you can conclude x > y without solving at all.

When signs alone don't settle it, factorise quickly, list both roots of each equation, and compare every root of x with every root of y. Only if one variable is greater in *all* comparisons can you conclude x > y. Otherwise the answer is "relationship cannot be determined" (CND) — which is the correct answer far more often than candidates think.

Example above: For x: 2x² − 7x + 6 = (2x − 3)(x − 2) → x = 1.5, 2. For y: 2y² − 9y + 10 = (2y − 5)(y − 2) → y = 2.5, 2. Comparing: sometimes x < y, sometimes x = y → CND.

🧠 Active Recall: In compare-x-and-y questions, when is the answer "relationship cannot be determined"?

Part 5: Series — Arithmetic, Geometric, and the Exam's Favourite Tricks

A series (or progression) is an ordered list of numbers following a rule.

Arithmetic Progression (AP): constant difference d between terms.

• nth term: aₙ = a + (n − 1)d
• Sum of n terms: Sₙ = n/2 × [2a + (n − 1)d] = n/2 × (first + last)

Geometric Progression (GP): constant ratio r between terms.

• nth term: aₙ = arⁿ⁻¹
• Sum of n terms: Sₙ = a(rⁿ − 1)/(r − 1) for r ≠ 1
• Infinite sum (|r| < 1): S∞ = a/(1 − r)

Number series questions in exams rarely announce themselves as AP or GP. Instead, look for these patterns, in this order:

1. Differences — are the gaps constant (AP), or do the gaps themselves form a pattern (×2, +3, squares…)?
2. Ratios — is each term a multiple of the previous (GP or mixed like ×2 + 1)?
3. Squares/cubes — terms like 4, 9, 16, 25 or 1, 8, 27, 64, possibly ± a small number.
4. Alternating or two interleaved series — odd positions follow one rule, even positions another.

Example: 3, 7, 15, 31, 63, ?
Differences: 4, 8, 16, 32 — doubling. Next difference 64 → 127. (Hidden rule: ×2 + 1.)

🧠 Active Recall: What are the four pattern checks for number series, in order? What is the infinite GP sum formula and when does it apply?

Part 6: Wrong Number Series — Find the Impostor

In this type, one term breaks the pattern. Strategy: first decode the rule from the *majority* of terms, then find the one that violates it.

Example: 2, 6, 12, 20, 32, 42
Differences: 4, 6, 8, 12, 10 — the pattern should be +4, +6, +8, +10, +12, giving 2, 6, 12, 20, 30, 42. So 32 is wrong (should be 30).

Notice the series is also n² + n: 2, 6, 12, 20, 30, 42. Recognising n² ± n patterns (2, 6, 12, 20, 30…) is a huge time-saver — memorise this family.

🧠 Active Recall: What is the n² + n series? How do you approach a wrong-number question differently from a missing-number one?

Summary Table

🔁 Review this after 2 days

Close the book and answer from memory:

1. Write the quadratic formula and the discriminant conditions.
2. Build the equation with roots 6 and −4.
3. Solve: 2, 5, 10, 17, 26, ? (Hint: n² + 1 → answer 37.)
4. In compare-x-and-y questions, when must you answer CND?

If any of these take more than 30 seconds, re-read that section — then review once more after a week.