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RRB NTPC Quadratic Equations

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This page covers RRB NTPC Quadratic Equations with complete concept notes, 39 graded practice MCQs, key points and exam-specific tips. Free to study.

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Concept Notes

Quadratic Equations— Rules & Concept

Core ConceptRead this first — the foundation of the topic

Quadratic Equations are polynomial equations with the highest degree of 2. They form the backbone of algebra questions in SSC CGL and appear in almost every exam paper. A quadratic equation has the standard form ax² + bx + c = 0, where 'a' cannot be zero. Core Properties and Rules:

Every quadratic equation has exactly two roots (solutions). These roots can be real and equal, real and unequal, or imaginary. The nature of roots depends on the discriminant (b² - 4ac). When discriminant > 0, roots are real and unequal. When discriminant = 0, roots are real and equal. When discriminant < 0, roots are imaginary.

Formula BlockMemorise — at least one formula appears in every paper
Quadratic Formula: x = [-b ± √(b² - 4ac)] / 2a
Sum of roots = -b/a
Product of roots = c/a
If roots are α and β, then equation is: x² - (α + β)x + αβ = 0
Exam PatternsWhat examiners ask — read before attempting PYQs

SSC CGL typically asks 2-3 questions on quadratic equations. Common question types include finding roots, determining nature of roots, forming equations from given roots, and word problems leading to quadratic equations. Questions often involve finding maximum/minimum values or solving practical problems.

ShortcutsUse these to save 30–60 seconds per question

#1 - Middle Term Splitting: For ax² + bx + c = 0, find two numbers whose product = ac and sum = b. Split the middle term using these numbers. This method is faster than the quadratic formula for most SSC questions. Shortcut Trick #2 - Perfect Square Recognition: If b² = 4ac, the equation is a perfect square.

The root is -b/2a (repeated twice). This saves calculation time.

Worked ExampleSolve this step-by-step before moving on
1
Step 1

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

2
Step 2

Find ac = 2 × 3 = 6

3
Step 3

Find two numbers whose product = 6 and sum = -7 These are -6 and -1 (since -6 × -1 = 6 and -6 + (-1) = -7)

4
Step 4

Rewrite: 2x² - 6x - x + 3 = 0

5
Step 5

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

6
Step 6

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

7
Step 7

Roots are x = 1/2 and x = 3 Worked Example 2: Find the equation whose roots are 3 and -2.

1
Step 1

Sum of roots = 3 + (-2) = 1

2
Step 2

Product of roots = 3 × (-2) = -6

3
Step 3

Using x² - (sum of roots)x + (product of roots) = 0

4
Step 4

Required equation: x² - x - 6 = 0 Shortcut Trick #3 - Vieta's Relations: Instead of finding individual roots, use sum and product formulas directly. This is especially useful when questions ask about expressions involving roots without finding the actual roots. Common Trap - The #1 Mistake: Students often forget to check if 'a' equals zero. If a = 0, the equation becomes linear (bx + c = 0), not quadratic. Always verify that the coefficient of x² is non-zero before applying quadratic methods. Another common error is sign mistakes while applying the quadratic formula, especially with the discriminant calculation. Exam Strategy: For SSC CGL, master middle term splitting first as it's faster than the quadratic formula. Practice identifying perfect squares quickly. Word problems often lead to quadratic equations, so focus on translation skills. Time management is crucial - spend maximum 2 minutes per quadratic equation question.

Key Points to Remember

  • Standard form: ax² + bx + c = 0 where a ≠ 0
  • Discriminant = b² - 4ac determines nature of roots
  • Sum of roots = -b/a, Product of roots = c/a
  • Middle term splitting is faster than quadratic formula for most SSC questions
  • If b² = 4ac, equation is perfect square with root = -b/2a
  • Every quadratic equation has exactly two roots
  • Roots can be real unequal, real equal, or imaginary
  • For forming equation from roots: x² - (sum)x + (product) = 0
  • Use Vieta's relations when questions involve expressions of roots
  • Always verify coefficient of x² is non-zero before applying quadratic methods

Exam-Specific Tips

  • Discriminant > 0 means real and unequal roots
  • Discriminant = 0 means real and equal roots
  • Discriminant < 0 means imaginary roots
  • Maximum value of quadratic expression ax² + bx + c occurs at x = -b/2a when a < 0
  • Minimum value of quadratic expression ax² + bx + c occurs at x = -b/2a when a > 0
  • If roots are reciprocals of each other, then a = c
  • If one root is negative of the other, then b = 0
  • Quadratic formula: x = [-b ± √(b² - 4ac)] / 2a
Practice MCQs

Quadratic Equations — Practice Questions

39graded MCQs · easy to hard · full solution & trap analysis · showing 20 of 39

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Practice 1easy

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

Practice 2easy

If the roots of x² - 6x + m = 0 are equal, find the value of m.

Practice 3easy

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

Practice 4easy

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

Practice 5easy

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

Practice 6easy

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

Practice 7easy

What is the product of the roots of the equation 3x² + 6x - 9 = 0?

Practice 8easy

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

Practice 9easy

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

Practice 10easy

If one root of the equation x² - 6x + k = 0 is 4, find the value of k.

Practice 11easy

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

Practice 12easy

If the sum of roots of the equation 3x² + (2m - 1)x + 5 = 0 is -2, find the value of m.

Practice 13easy

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

Practice 14easy

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

Practice 15medium

If the roots of the quadratic equation x² − 5x + k = 0 are in the ratio 2:3, find the value of k.

Practice 16medium

If the roots of the quadratic equation x² - 10x + k = 0 are in the ratio 2:3, find the value of k.

Practice 17medium

The roots of the equation 2x² - 8x + k = 0 are reciprocals of each other. Find k.

Practice 18medium

The sum of the squares of two consecutive integers is 145. Which quadratic equation represents this condition?

Practice 19medium

If α and β are roots of x² - 5x + 6 = 0, find the value of (α - β)².

Practice 20medium

A quadratic equation has roots that differ by 5. If the sum of the roots is 7, what is the equation?

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60-Second Revision — Quadratic Equations

  • Formula: Sum = -b/a, Product = c/a, Discriminant = b² - 4ac
  • Remember: Use middle term splitting for faster solutions in SSC
  • Trap: Always check if coefficient of x² is non-zero
  • Shortcut: If b² = 4ac, root is -b/2a (perfect square)
  • Pattern: 2-3 questions appear in every SSC CGL paper
  • Strategy: Master Vieta's relations for expressions involving roots
  • Time limit: Maximum 2 minutes per quadratic equation question
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