Quadratic Equation Questions

Quadratic equation questions are a fundamental part of algebra and appear regularly in exams and practical applications. These questions require a solid grasp of the quadratic equation formula, the nature of roots, and how to approach different solution methods. In this article, we’ll explore definitions, formulas, types of roots, graphs, and multiple solved examples to sharpen your problem-solving skills.

 

Table of Contents

• Definition
• Quadratic Equation Formula
• Nature of Roots
• Sum and Product of Roots
• Graphical Interpretation
• Common Question Types
• Solved Quadratic Equation Questions
• Tips to Solve Faster
• FAQs on Quadratic Equation Questions

 

Quadratic Equation Definition

A quadratic equation is a second-degree polynomial equation in one variable, generally represented as:

ax² + bx + c = 0

where a ≠ 0 and a, b, c are real numbers. The values of x that satisfy the equation are called the roots of the equation.

 

Quadratic Equation Formula

To solve for x, we use the quadratic formula:

x = (-b ± √(b² - 4ac)) / (2a)

This formula is derived using the method of completing the square and works for all types of quadratic equations.

 

Nature of Roots

 The discriminant D = b² - 4ac determines the nature of the roots:

• D > 0: Two real and distinct roots

• D = 0: Two real and equal roots

• D < 0: Two complex roots

 

Sum and Product of Roots

 Let α and β be the roots of the equation:

• Sum of roots: α + β = -b/a

• Product of roots: αβ = c/a

This is useful when you're asked to find an equation from its roots.

 

Graphical Interpretation

 The graph of a quadratic equation is a parabola:

• Opens upward if a > 0

• Opens downward if a < 0

• The vertex of the parabola lies at x = -b/2a

• The discriminant indicates if the parabola touches, intersects, or doesn’t meet the x-axis

 

Common Question Types

• Find the roots using the quadratic formula

• Identify the nature of the roots

• Frame equations given sum and product of roots

• Determine range of values for a condition on roots

• Solve real-life application-based quadratic problems

 

Solved Quadratic Equation Questions with Solutions

Q1. Solve the quadratic equation:
2x² - 7x + 3 = 0

Solution:
a = 2, b = -7, c = 3
D = (-7)² - 4×2×3 = 49 - 24 = 25
x = (7 ± √25)/4 = (7 ± 5)/4
x = 3, x = 0.5

 

Q2. Find the value of k for which the roots of x² + 4kx + 8 = 0 are equal.

Solution:
D = (4k)² - 4×1×8 = 16k² - 32
Set D = 0:
16k² - 32 = 0 → k² = 2 → k = ±√2

 

Q3. Form a quadratic equation whose roots are 5 and 7.

Solution:
Sum = 5 + 7 = 12, Product = 5×7 = 35
Equation: x² - 12x + 35 = 0

 

Q4. Determine the nature of roots of 4x² + 4x + 1 = 0

Solution:
D = 4² - 4×4×1 = 16 - 16 = 0
So, roots are real and equal.

 

Q5. For what values of k does x² - (k - 2)x + k + 4 = 0 have real roots?

Solution:
D = (k - 2)² - 4(k + 4) ≥ 0
→ k² - 8k - 12 ≥ 0
→ k ∈ (-∞, 2 - 2√4) ∪ (2 + 2√4, ∞)

 

Tips to Solve Faster

• Always compute the discriminant first

• Use sum and product of roots for quick formulation

• Factorisation is faster when roots are integers

• Use graphs to visualise root nature

 

Related Articles : 

Factorization of a Quadratic Equation: Learn the easiest tricks to factor quadratic equations - visit our blog at Orchids The International School now!

 

FAQs on Quadratic Equation Questions

Q1. What is the standard quadratic equation formula?

Ans: x = (-b ± √(b² - 4ac)) / (2a)

 

Q2. How do I know the nature of roots?

Ans: Check the discriminant D:

• D > 0: real and unequal

• D = 0: real and equal

• D < 0: complex

 

Q3. How can I quickly form a quadratic equation from given roots?

Ans:  If roots are p and q, then the equation is x² - (p + q)x + pq = 0

 

Q4. Are quadratic equations used in real life?

Ans: Yes. They're used in physics (projectile motion), business (profit curves), engineering (parabolic structures), and more.

 

Q5. What’s the easiest way to solve quadratic equations?

Ans: Use factorisation when possible; else apply the quadratic formula.

 

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