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CDS Quadratic Equations & Polynomials

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This page covers CDS Quadratic Equations & Polynomials with complete concept notes, 36 graded practice MCQs, key points and exam-specific tips. Free to study.

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Concept Notes

Quadratic Equations & Polynomials— Rules & Concept

1.Core Concept

Read this first — the foundation of the topic

A quadratic equation is any equation that can be written in the form ax² + bx + c = 0, where a, b, c are constants and a ≠ 0. These equations form the backbone of algebra and appear in almost every NDA math paper.

2.Key Rules

Core rules you must know cold

Every quadratic equation has exactly 2 roots (solutions)

Roots can be real or imaginary

If discriminant > 0: two distinct real roots

If discriminant = 0: one repeated real root

If discriminant < 0: two imaginary roots

3.Formula Block

Memorise — at least one formula appears in every paper

• Standard form: ax² + bx + c = 0

• Discriminant: D = b² - 4ac

• Quadratic formula: x = (-b ± √D)/2a

• If roots are α, β: equation is x² - (α+β)x + αβ = 0

• Sum of roots: -b/a

• Product of roots: c/a

4.Exam Patterns

What examiners ask — read before attempting PYQs

Finding roots using factorization or formula

Nature of roots using discriminant

Sum and product of roots problems

Word problems leading to quadratic equations

Polynomial remainder theorem applications

5.Shortcuts

Use these to save 30–60 seconds per question

For equations like x² ± px ± q = 0, try factors of 'q' first. This saves time over using the quadratic formula.

6.Worked Example

Solve this step-by-step before moving on

1

Step 1

Identify a=2, b=-7, c=3

2

Step 2

Try factorization first: Look for factors of (2×3=6) that add to -7

3

Step 3

Factors are -6 and -1: 2x² - 6x - x + 3 = 0

4

Step 4

Group: 2x(x-3) - 1(x-3) = 0

5

Step 5

Factor: (2x-1)(x-3) = 0

6

Step 6

Solutions: x = 1/2 or x = 3 Worked Example 2: If roots of x² - 5x + k = 0 are in ratio 2:3, find k.

1

Step 1

Let roots be 2m and 3m

2

Step 2

Sum of roots: 2m + 3m = 5 (coefficient rule)

3

Step 3

So 5m = 5, therefore m = 1

4

Step 4

Roots are 2 and 3

5

Step 5

Product of roots: 2 × 3 = k = 6 Mega Shortcut - Vieta's Relations: Instead of solving complex quadratics, use sum = -b/a and product = c/a directly. This works for 60% of NDA questions. Speed Trick - Perfect Square Check: For ax² + bx + c, if b² = 4ac, then it's a perfect square: (√ax ± √c)² The #1 Most Common Trap: Students forget that a ≠ 0 in quadratic equations. When solving word problems, always check if your equation is actually quadratic. Also, many students confuse the signs in Vieta's formulas - remember sum of roots is -b/a (negative coefficient ratio). Polynomial Connection: Quadratic equations are second-degree polynomials. The remainder theorem states that when polynomial P(x) is divided by (x-a), remainder = P(a). This concept frequently combines with quadratic equations in NDA papers.

Key Points to Remember

Standard form: ax² + bx + c = 0 where a ≠ 0
Discriminant D = b² - 4ac determines nature of roots
Sum of roots = -b/a, Product of roots = c/a (Vieta's formulas)
Quick factorization: try factors of 'ac' that add to 'b'
If D > 0: two distinct real roots, D = 0: equal roots, D < 0: imaginary roots
Quadratic formula: x = (-b ± √D)/2a works for all cases
Speed check: if b² = 4ac, equation is a perfect square
For ratio problems: let roots be rm and sn, use sum/product relations
Every quadratic has exactly 2 roots (counting multiplicity)
Formation shortcut: if roots are p,q then equation is x² - (p+q)x + pq = 0

Exam-Specific Tips

Discriminant formula is exactly b² - 4ac, not 4ac - b²
Sum of roots of ax² + bx + c = 0 is always -b/a (negative ratio)
Product of roots of ax² + bx + c = 0 is always c/a
Maximum value of ax² + bx + c occurs at x = -b/2a when a < 0
A quadratic equation ax² + bx + c = 0 has equal roots when b² = 4ac exactly
The coefficient 'a' in ax² + bx + c = 0 must be non-zero for it to be quadratic
Remainder when polynomial P(x) is divided by (x-a) equals P(a)
If one root of x² + px + q = 0 is reciprocal of other, then q = 1

Practice MCQs

Quadratic Equations & Polynomials — Practice Questions

36graded MCQs · easy to hard · full solution & trap analysis · showing 20 of 36

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Practice 1easy

If one root of the equation 2x² − 3x + k = 0 is twice the other root, then the value of k is:

Practice 2easy

The polynomial P(x) = x³ − 6x² + 11x − 6 has roots in arithmetic progression. If the roots are (a − d), a, and (a + d), then the value of a is:

Practice 3easy

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

Practice 4easy

If one root of the quadratic equation 2x² + 5x + m = 0 is the reciprocal of the other, then m equals:

Practice 5easy

If α and β are the roots of the quadratic equation x² − 5x + 6 = 0, then the value of α³ + β³ is:

Practice 6easy

The quadratic equation x² + px + q = 0 has roots that are reciprocals of each other. Which of the following must be true?

Practice 7easy

If the roots of x² − 6x + k = 0 are real and distinct, then the range of k is:

Practice 8easy

The polynomial P(x) = x³ − 7x² + 14x − 8 has a root at x = 1. What is the sum of the other two roots?

Practice 9easy

If one root of the equation 2x² − 5x + c = 0 is twice the other, then the value of c is:

Practice 10easy

The quadratic equation x² + px + q = 0 has roots that are reciprocals of each other. Which of the following is necessarily true?

Practice 11easy

If the polynomial P(x) = x³ − 6x² + 11x − 6 has roots α, β, and γ, then α + β + γ equals:

Practice 12easy

The quadratic equation x² − 2kx + (k² − 1) = 0 has equal roots. The value of k is:

Practice 13medium

If the roots of the equation x² + px + q = 0 are in the ratio 2:3, then which of the following is correct?

Practice 14medium

The polynomial P(x) = x³ - 6x² + 11x - 6 has roots α, β, and γ. If one root is 1, then the sum of the other two roots is:

Practice 15medium

If the roots of x² + px + q = 0 are in the ratio 2:3, and their sum is 10, then the value of q is:

Practice 16medium

The quadratic equation x² - (k+1)x + (k-1) = 0 has real roots for all real values of k if and only if:

Practice 17medium

If the quadratic equation x² - (k + 1)x + (k² - 3k + 2) = 0 has equal roots, then the value of k is:

Practice 18medium

If α and β are the roots of x² + bx + c = 0, and γ and δ are the roots of x² + px + q = 0, and if α + γ = β + δ = -1, then which of the following is correct?