Quadratic Equations is one of the most important chapters in Class 10 Mathematics. Every year, questions from this chapter appear in the CBSE board exams as well as in other competitive exams such as NTSE, Olympiads, and even foundation-level JEE preparation. A solid understanding of quadratic equations not only helps in scoring good marks but also strengthens your base for higher mathematics.

In this article, we will cover the complete NCERT Class 10 Quadratic Equations chapter in a simple and easy-to-understand way. We will discuss definitions, methods of solving, applications, formulas, and important exam questions.

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What is a Quadratic Equation?

A quadratic equation is an equation of the form:

ax² + bx + c = 0, where a ≠ 0

Here:

• a, b, and c are real numbers.

• a is the coefficient of x², b is the coefficient of x, and c is the constant term.

For example:

• 2x² + 3x – 5 = 0

• x² – 7x + 12 = 0

Both are quadratic equations.

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Standard Forms and Roots of Quadratic Equations

The solutions of quadratic equations are called roots. They can be found by several methods:

1. Factorisation method – Splitting the middle term.

2. Completing the square method.

3. Quadratic formula method.

The quadratic formula is:

x = (-b ± √(b² – 4ac)) / 2a

The term b² – 4ac is called the discriminant (D).

• If D > 0 → Two distinct real roots.

• If D = 0 → Two equal real roots.

• If D < 0 → No real roots.

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Methods of Solving Quadratic Equations – Step by Step

1. Solving by Factorisation

Example: Solve x² – 7x + 12 = 0
Step 1: Break middle term – x² – 3x – 4x + 12 = 0
Step 2: Group terms – x(x – 3) – 4(x – 3) = 0
Step 3: Factorise – (x – 3)(x – 4) = 0
Roots = 3, 4

2. Solving by Completing the Square

Example: Solve x² + 6x – 7 = 0
Step 1: x² + 6x = 7
Step 2: Add (b/2)² = (6/2)² = 9 → x² + 6x + 9 = 7 + 9
Step 3: (x + 3)² = 16
Step 4: x + 3 = ±4
Roots = 1, –7

3. Solving by Quadratic Formula

Example: Solve 2x² – 3x – 2 = 0
Here a = 2, b = –3, c = –2
D = (–3)² – 4(2)(–2) = 9 + 16 = 25
x = [3 ± √25] / 4 = [3 ± 5] / 4
Roots = 2, –½

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Applications of Quadratic Equations

Quadratic equations are not only mathematical concepts but also practical tools used in real life. Some common applications include:

• Calculating areas (geometry-based problems).

• Finding speed, time, and distance in motion-related problems.

• Business and profit-loss problems.

• Physics equations of motion.

• Height and distance problems.

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Important Formulas for Quadratic Equations

• General form: ax² + bx + c = 0

• Roots by quadratic formula: x = (–b ± √(b² – 4ac)) / 2a

• Nature of roots depends on D = b² – 4ac

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Tips to Score High Marks in Quadratic Equations

1. Practice factorisation regularly as it saves time.

2. Always check discriminant before applying formulas.

3. Write step-by-step solutions in exams.

4. Memorise important formulas.

5. Attempt NCERT solved examples and exercises.

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Important Questions from NCERT Quadratic Equations

1. Solve: 2x² – 5x + 3 = 0

2. The product of two consecutive positive integers is 182. Find the integers.

3. A train travels 120 km at a uniform speed. If the speed had been 10 km/hr more, it would have taken 30 minutes less. Find the speed of the train.

4. The sum of the reciprocals of Rehman’s ages (in years) 3 years ago and 5 years from now is 1/3. Find his present age.

5. If the sum of the areas of two squares is 468 m² and the difference of their perimeters is 24 m, find the sides of the two squares.

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People Also Ask

What is the easiest method to solve quadratic equations?

Factorisation is the simplest method when the equation is factorisable. Otherwise, the quadratic formula works for all cases.

How many questions come from quadratic equations in Class 10 board exam?

Generally, 3 to 5 marks questions are asked, and sometimes application-based word problems are included.

What are real-life uses of quadratic equations?

They are used in physics (projectile motion), engineering, business, and economics.

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How to Prepare Quadratic Equations for Board Exams

1. Start with NCERT solved examples.

2. Solve all exercise questions.

3. Practice additional problems from sample papers.

4. Revise important formulas and discriminant cases.

5. Take timed practice tests.

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FAQs on Quadratic Equations Class 10

Q1. What is the formula for solving quadratic equations?
Ans. The quadratic formula is x = (–b ± √(b² – 4ac)) / 2a.

Q2. How do I know if a quadratic equation has real roots?
Ans. If the discriminant (b² – 4ac) is greater than or equal to zero, the equation has real roots.

Q3. Is quadratic equations important for Class 10 boards?
Ans. Yes, it is a high-weightage chapter and is also useful for higher studies.

Q4. Which is better: factorisation or quadratic formula?
Ans. Factorisation is quick but works only for simple numbers. The quadratic formula works in all cases.