Quadratic Equations Class 10 Most Important Question And MCQ

Quadratic Equations Class 10 Complete Notes | Explanation, MCQ, HOTS, PYQ & Practice Questions

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Introduction ( Quadratic Equations Class 10 )

Quadratic Equations is one of the most important chapters in Class 10 Mathematics. It forms the foundation for higher mathematics and is frequently asked in board examinations, competitive exams, and scholarship tests.

A quadratic equation is an equation of degree 2, meaning the highest power of the variable is 2. In this chapter, students learn how to solve quadratic equations using different methods such as factorization, completing the square, and the quadratic formula.

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विषय परिचय (Hindi) ( Quadratic Equations Class 10 )

द्विघात समीकरण (Quadratic Equations) कक्षा 10 गणित का अत्यंत महत्वपूर्ण अध्याय है। बोर्ड परीक्षा, NTSE, ओलंपियाड तथा विभिन्न प्रतियोगी परीक्षाओं में इससे नियमित प्रश्न पूछे जाते हैं।

जिस समीकरण में चर (Variable) की उच्चतम घात 2 हो, उसे द्विघात समीकरण कहते हैं।

उदाहरण:$$x^2+5x+6=0$$x2+5x+6=0

यह एक Quadratic Equation है क्योंकि इसमें x की उच्चतम घात 2 है।

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Table of Contents ( Quadratic Equations Class 10 )

1. Introduction to Quadratic Equations
2. Definition of Quadratic Equation
3. Standard Form
4. Important Terms
5. Methods of Solving Quadratic Equations
6. Factorization Method
7. Completing Square Method
8. Quadratic Formula Method
9. Nature of Roots
10. Discriminant
11. Real-Life Applications
12. Key Points
13. MCQs
14. HOTS Questions
15. PYQs
16. Expected Questions
17. FAQs
18. Viewer Challenge Questions

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Abbreviations ( Quadratic Equations Class 10 )

Abbreviation	Full Form
QE	Quadratic Equation
D	Discriminant
RHS	Right Hand Side
LHS	Left Hand Side
MCQ	Multiple Choice Question
HOTS	Higher Order Thinking Skills
PYQ	Previous Year Question
CBSE	Central Board of Secondary Education
NCERT	National Council of Educational Research and Training

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What is a Quadratic Equation?

A polynomial equation of degree 2 is called a Quadratic Equation.

Standard Form

$$ax^2+bx+c=0$$ax2+bx+c=0

Where:

• a ≠ 0
• a, b, c are real numbers

Examples

• $x^2+4x+3=0$
• $5x^2+9=0$

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द्विघात समीकरण की परिभाषा

यदि किसी समीकरण का सामान्य रूप$$ax^2+bx+c=0$$ax2+bx+c=0

हो तथा $a \neq 0$, तो उसे द्विघात समीकरण कहते हैं।

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Important Terms

Variable

जिसका मान बदल सकता है।

उदाहरण: x

Coefficient

Variable के साथ जुड़ी संख्या।

उदाहरण:$$3x^2$$

यहाँ 3 coefficient है।

Constant

नियत संख्या।

उदाहरण:$$x^2+4x+7=0$$

यहाँ 7 constant है।

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Methods of Solving Quadratic Equations

There are three major methods:

1. Factorization Method

2. Completing the Square Method

3. Quadratic Formula Method

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1. Factorization Method

Example

Solve:$$x^2+5x+6=0$$

Step 1:

Find two numbers whose sum = 5 and product = 6

Numbers = 2 and 3

Step 2:$$x^2+2x+3x+6=0$$

Step 3:$$x(x+2)+3(x+2)=0$$$$(x+2)(x+3)=0$$

Therefore,$$x=-2,\,-3$$x=−2,−3

Answer

Roots = -2 and -3

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2. Completing the Square Method

Example

Solve:$$x^2+6x+5=0$$$$x^2+6x=-5$$

Add 9 on both sides$$x^2+6x+9=4$$$$(x+3)^2=4$$$$x+3=\pm2$$$$x=-1,-5$$

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3. Quadratic Formula Method

Formula:$$x=\frac{-b\pm\sqrt{b^2-4ac}}{2a}$$

Example

$$2x^2-5x+2=0$$

Here

a = 2

b = -5

c = 2$$x=\frac{5\pm\sqrt{25-16}}{4}$$$$x=\frac{5\pm3}{4}$$

Roots:$$x=2,\frac12$$

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Discriminant

The expression$$D=b^2-4ac$$

is called the Discriminant.

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Nature of Roots

Case 1

D > 0

Roots are real and distinct.

Example:$$x^2-5x+6=0$$

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Case 2

D = 0

Roots are real and equal.

Example:$$x^2-4x+4=0$$

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Case 3

D < 0

Roots are imaginary.

Example:$$x^2+4x+8=0$$

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Real-Life Applications

Quadratic equations are used in:

• Projectile Motion
• Engineering
• Architecture
• Sports Analytics
• Business Calculations
• Construction Work
• Physics Problems

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Key Points (Quick Revision)

✅ Degree must be 2

Standard Form:$$ax^2+bx+c=0$$

✅ a cannot be zero

Discriminant:$$D=b^2-4ac$$

✅ Quadratic Formula:$$x=\frac{-b\pm\sqrt{b^2-4ac}}{2a}$$

✅ Nature of roots depends upon D

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MCQs (Important)

Q1

The degree of a quadratic equation is:

A. 1

B. 2

C. 3

D. 4

Answer: B

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Q2

Standard form is:

A. ax+b=0

B. ax²+bx+c=0

C. ax³+bx+c=0

D. ax⁴+bx+c=0

Answer: B

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Q3

Discriminant equals:

A. b²+4ac

B. b²−4ac

C. a²+b²

D. c²−4ab

Answer: B

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Q4

Roots of x²−9=0 are:

A. ±3

B. ±9

C. 3

D. 9

Answer: A

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Q5

If D = 0, roots are:

A. Distinct

B. Imaginary

C. Equal

D. Irrational

Answer: C

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HOTS Questions

Q1

If one root of$$x^2-7x+k=0$$

is 3, find k.

Solution

3²−7(3)+k=0

9−21+k=0

k=12

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Q2

Find a quadratic equation whose roots are 4 and −3.

Solution:

Sum = 1

Product = −12

Equation:$$x^2-x-12=0$$

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Q3

A rectangular garden has area 96 m². Length exceeds breadth by 4 m. Find dimensions.

Answer:

Breadth = 8 m

Length = 12 m

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Previous Year Questions (PYQ)

Q1 (CBSE)

Solve:$$x^2-3x-10=0$$

Answer:$$(x-5)(x+2)=0$$

Roots = 5, -2

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Q2

Solve by quadratic formula:$$3x^2-2x-1=0$$

Answer:$$x=1,\,-\frac13$$

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Q3

Find nature of roots:$$x^2-6x+9=0$$

D=0

Roots are equal.

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Q4

Find roots:$$2x^2+7x+3=0$$

Answer:$$x=-3,\,-\frac12$$

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Expected Board Examination Questions

Short Answer

1. Define quadratic equation.
2. Write standard form.
3. What is discriminant?
4. State quadratic formula.
5. What are equal roots?

Long Answer

1. Solve by (a) factorization.
   • (b) completing square.
     • (c)quadratic formula.
2. Determine nature of roots.
3. Form quadratic equation from given roots.

Case-Based Questions

1. Area and perimeter problems.
2. Speed and distance problems.
3. Rectangular field questions.
4. Profit-loss application problems.

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Frequently Asked Questions (FAQ)

Q1. What is a quadratic equation?

An equation whose highest power of variable is 2.

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Q2. Can a be zero?

No. If a = 0, the equation becomes linear.

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Q3. What is discriminant?

$$D=b^2-4ac$$

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Q4. Which method is easiest?

Factorization is easiest when factors are easily available.

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Q5. Why is quadratic formula important?

It works for every quadratic equation.

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Q6. How many roots can a quadratic equation have?

Maximum two roots.

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Q7. Is this chapter important for boards?

Yes. Almost every year questions are asked from this chapter.

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Exam Tips

Learn quadratic formula perfectly.

Practice factorization daily.

Remember discriminant cases.

Solve NCERT examples multiple times.

Practice PYQs of last 10 years.

Focus on word problems.

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Viewer Challenge Section

🎯 Solve these questions and post your answers in the comments or mail us through www.risingstarmindset.com

Question 1

Solve:$$x^2-8x+15=0$$x2−8x+15=0

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Question 2

Find the roots of:$$2x^2+5x-3=0$$2×2+5x−3=0

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Question 3

Determine the nature of roots:$$x^2+2x+5=0$$x2+2x+5=0

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Question 4

Form a quadratic equation whose roots are 6 and -2.

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Question 5

The product of two consecutive integers is 156. Find the integers using quadratic equations.

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Special Announcement for Viewers

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