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SSC CGL Quadratic Equations

Study Material — 2 PYQs (2018–2020) · Concept Notes · Shortcuts

SSC CGL Quadratic Equations is a frequently tested subtopic — 2 previous year questions from 2018–2020 papers are included below with concept notes, key rules and shortcut tricks.

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2018–2020

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Previous Year Questions

SSC CGL Quadratic Equations — Past Exam Questions

2 questions from actual SSC CGL papers · all shown free · click option to reveal solution

Exam Q 12018Previous Year Pattern

Find the roots of the quadratic equation x² - 5x + 6 = 0.

Exam Q 22020Previous Year Pattern

If the roots of the quadratic equation x² − (p + q)x + pq = 0 are α and β, and α − β = 6, then which of the following is true?

Concept Notes

Quadratic Equations— Rules & Concept

Core ConceptRead this first — the foundation of the topic

QUADRATIC EQUATIONS — COMPLETE GUIDE FOR SSC CGL ━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━

CORE CONCEPT ━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━

A quadratic equation is any equation where the highest power of x is 2. It always looks like this: ax² + bx + c = 0

Here, a, b, c are numbers (called coefficients), and a cannot be zero. If a = 0, it becomes a simple linear equation — not quadratic. Every quadratic equation has exactly TWO roots (solutions). These roots tell you the values of x that satisfy the equation. The roots are often written as α (alpha) and β (beta).

━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━

Key RulesCore rules you must know cold

━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━ These two formulas are the backbone of almost all SSC questions: Sum of roots: α + β = -b/a Product of roots: α × β = c/a Nature of roots depends on the Discriminant (D): D = b² - 4ac If D > 0 → Two distinct real roots If D = 0 → Two equal (same) real roots If D < 0 → No real roots (imaginary roots) ━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━

Formula BlockMemorise — at least one formula appears in every paper

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Quadratic Formula (use when factoring is hard):

x = [-b ± √(b² - 4ac)] / 2a

Building equation from roots:

x² - (Sum of roots)x + (Product of roots) = 0

x² - (α + β)x + αβ = 0

Key identity shortcuts:

α² + β² = (α + β)² - 2αβ

α² - β² = (α + β)(α - β)

α³ + β³ = (α + β)³ - 3αβ(α + β)

(α - β)² = (α + β)² - 4αβ

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Exam PatternsWhat examiners ask — read before attempting PYQs

TRICK 2 — Roots with opposite sign

If roots are -α and -β, replace x with -x in original equation

TRICK 3 — Equal roots shortcut

When roots are equal, D = 0, so b² = 4ac. Use this to find unknown k in seconds without solving the full equation. ━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━

Worked ExampleSolve this step-by-step before moving on

1

Step 1

Identify a=2, b=-5, c=3

2

Step 2

Sum of roots = α + β = -(-5)/2 = 5/2

3

Step 3

Product of roots = αβ = 3/2

4

Step 4

Use identity → α² + β² = (α + β)² - 2αβ = (5/2)² - 2(3/2) = 25/4 - 3 = 25/4 - 12/4 = 13/4 Answer: 13/4 ━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━ WORKED EXAMPLE 2 (Forming New Equation) ━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━ Q: Roots of x² - 6x + 8 = 0 are α and β. Form equation with roots 1/α and 1/β.

1

Step 1

a=1, b=-6, c=8

2

Step 2

New sum = 1/α + 1/β = (α + β)/αβ = 6/8 = 3/4

3

Step 3

New product = 1/α × 1/β = 1/αβ = 1/8

4

Step 4

New equation = x² - (3/4)x + 1/8 = 0 Multiply throughout by 8 → 8x² - 6x + 1 = 0 SHORTCUT CHECK: Just swap a and c → 8x² - 6x + 1 = 0. Same answer! Trick works perfectly. ━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━ #1 MOST COMMON TRAP — STUDENTS LOSE MARKS HERE ━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━ When finding sum of roots, the formula is -b/a, NOT b/a. For 2x² - 5x + 3 = 0, b = -5. So sum = -(-5)/2 = +5/2. Students forget the negative sign and write 5/2 for b directly — they get -5/2 instead. This one sign error changes the entire answer. Always re-check the sign of b before substituting.

Key Points to Remember

Standard form of quadratic equation: ax² + bx + c = 0, where a ≠ 0
Every quadratic equation has exactly TWO roots (real or imaginary)
Formula: Sum of roots α + β = -b/a (note the negative sign)
Formula: Product of roots α × β = c/a
Discriminant D = b² - 4ac determines nature of roots
If D > 0 → two distinct real roots; D = 0 → two equal roots; D < 0 → no real roots
Shortcut: α² + β² = (α + β)² - 2αβ — use this identity to avoid full solving
Shortcut: Equation with reciprocal roots (1/α, 1/β) → just swap coefficients a and c
Shortcut: For equal roots condition, set D = 0 i.e. b² = 4ac to find unknown k
Form equation from roots using: x² - (Sum)x + (Product) = 0

Exam-Specific Tips

In ax² + bx + c = 0, the sum of roots is -b/a and product of roots is c/a — these two formulas are called Vieta's Formulas
Discriminant formula: D = b² - 4ac; when D = 0, both roots are equal and real
The quadratic formula is x = [-b ± √(b² - 4ac)] / 2a
If roots are α and β, then α² + β² = (α + β)² - 2αβ and α³ + β³ = (α + β)³ - 3αβ(α + β)
For equation with roots 1/α and 1/β: replace original ax² + bx + c = 0 with cx² + bx + a = 0 (swap a and c)
If both roots are positive: sum > 0 and product > 0; if both roots are negative: sum < 0 and product > 0
If roots have opposite signs: product of roots is negative (c/a < 0)
(α - β)² = (α + β)² - 4αβ — this identity is used to find difference of roots without solving the equation

Practice MCQs

Quadratic Equations — Practice Questions

38graded MCQs · easy to hard · full solution & trap analysis · showing 20 of 38

All MCQs →

Practice 1easy

What is the product of the roots of the equation 2x² - 8x + 6 = 0?

Practice 2easy

Solve: x² - 5x + 6 = 0. What are the roots?

Practice 3easy

If the roots of x² + px + 12 = 0 are 3 and 4, find the value of p.

Practice 4easy

Find the product of the roots of the equation 2x² - 8x + 6 = 0.

Practice 5easy

The quadratic equation x² + 4x + 4 = 0 has roots. What is the nature of these roots?

Practice 6easy

If one root of the equation x² - 6x + k = 0 is 2, find the value of k.

Practice 7easy

Which of the following is a root of x² - 6x + 8 = 0?

Practice 8easy

If one root of the equation x² - 5x + k = 0 is 2, find the value of k.

Practice 9easy

If one root of the equation x² - 5x + 6 = 0 is 2, find the other root.

Practice 10easy

If one root of the equation x² - 5x + k = 0 is 2, find the value of k.

Practice 11easy

If the roots of x² + bx + 20 = 0 are 4 and 5, find the value of b.

Practice 12easy

The product of the roots of the equation 3x² + 6x - 9 = 0 is:

Practice 13easy

If x² - 7x + 12 = 0, find the sum of the roots of the equation.

Practice 14easy

What is the product of the roots of the equation 2x² - 8x + 6 = 0?

Practice 15easy

If the roots of the equation 2x² - 8x + k = 0 are equal, find the value of k.

Practice 16easy

Solve: x² - 9 = 0. What are the roots?

Practice 17easy

If x² - 7x + 12 = 0, find the sum of the roots of the equation.

Practice 18medium

If the roots of the quadratic equation x² − (p + q)x + pq = 0 are α and β, and α − β = 6, then find the value of (α + β)² − 4αβ in terms of the roots.

Practice 19medium

If one root of the equation 2x² - 5x + c = 0 is 2, find the other root and the value of c.

Practice 20medium

If the roots of the quadratic equation x² + px + q = 0 are 3 and 5, then what is the value of p² - 4q?

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60-Second Revision — Quadratic Equations

Formula: Sum of roots = -b/a and Product of roots = c/a — memorise with sign, never drop the negative
Formula: D = b² - 4ac — D > 0 (two real roots), D = 0 (equal roots), D < 0 (no real roots)
Shortcut: α² + β² = (α + β)² - 2αβ — most asked identity; compute in one step from sum and product
Shortcut: Reciprocal roots equation → just swap a and c in original equation, keep b same
Trap: b in the equation 2x² - 5x + 3 is -5, not 5 — always pick up the sign of b carefully
Remember: To form equation from roots → x² - (Sum)x + (Product) = 0
Shortcut: Equal roots condition → b² = 4ac; use this instantly to find unknown k without quadratic formula

←Linear Equations Algebraic Identities→

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