‘रचना-प्रयोग’ का मुख्य उद्देश्य क्या है?

व्याकरण का व्यावहारिक प्रयोग सिखाना

‘द्वितीयः’ किस प्रकार की संख्या है?

क्रमवाचक (दूसरा)

What does the discriminant tell us about the roots?

If \(b^2 - 4ac > 0\) the roots are real and distinct; if \(= 0\) they are real and equal; if \(< 0\) there are no real roots.

Solve \(x^2 - 5x + 6 = 0\) by factorisation.

\((x-2)(x-3) = 0\), so \(x = 2\) or \(x = 3\).

What is the standard form of a quadratic equation?

\(ax^2 + bx + c = 0\), where \(a, b\) and \(c\) are real numbers and \(a \ne 0\).

Name the methods of solving a quadratic equation.

By factorisation, by completing the square, and by using the quadratic formula.

Write the quadratic formula.

For \(ax^2 + bx + c = 0\), \(x = \dfrac{-b \pm \sqrt{b^2 - 4ac}}{2a}\).

What is the discriminant, and what does it tell us?

The discriminant is \(D = b^2 - 4ac\); if \(D > 0\) the roots are real and distinct, if \(D = 0\) real and equal, and if \(D < 0\) there are no real roots.

Solve \(x^2 - 5x + 6 = 0\) by factorisation.

\((x - 2)(x - 3) = 0\), so \(x = 2\) or \(x = 3\).

Find the nature of the roots of \(2x^2 - 3x + 5 = 0\).

\(D = (-3)^2 - 4(2)(5) = 9 - 40 = -31 < 0\), so the equation has no real roots.

For what value of \(k\) does \(x^2 + kx + 4 = 0\) have equal roots?

Equal roots need \(D = 0\): \(k^2 - 16 = 0\), so \(k = \pm 4\).

The product of two consecutive positive integers is 56. Frame and solve the equation.

Let the integers be \(x\) and \(x + 1\): \(x(x + 1) = 56 \Rightarrow x^2 + x - 56 = 0 \Rightarrow (x + 8)(x - 7) = 0\), so \(x = 7\) and the integers are 7 and 8.

CBSE · Board years

CBSE Class 10 — Notes, Chapters & Practice Quizzes

Your Class 10 year, organised — chapter notes, board-style questions and a practice quiz for every chapter.

7 subjects 143 chapters 149 practice quizzes NCERT syllabus

126 quiz attempts so far Dual AI-verified questions 10 quizzes free, then ₹9 per quiz

Notice Board

10 questions · instant score · weak-chapter diagnosis

01 Choose a chapter02 Revise key ideas03 Test recall

Students preparing for CBSE Class 10 Board Examination

View all

Formula of the day

Maths · Quadratic equations · 9 formulae in the cheatsheet

Trap of the day

sin²θ means (sinθ)², not sin(θ²). A pure notation trap that changes the value.

IIT Guidance Practice QuizFind weak topics English Health and Physical Education Hindi Mathematics Sanskrit Science PYQ PapersSolve official papers Mock TestsRehearse exam day DiscussionAsk the community

Mathematics · 14 chapters

Summary, key terms, extra questions with answers and a practice quiz with AI diagnosis for each.

Chapter 4: Quadratic EquationsClass 10 Mathematics — summary, notes, extra questions & NCERT solutions

Q: Solve \(x^2 - 5x + 6 = 0\) by factorisation.

\((x-2)(x-3) = 0\), so \(x = 2\) or \(x = 3\).

Q: What is the standard form of a quadratic equation?

\(ax^2 + bx + c = 0\), where \(a, b\) and \(c\) are real numbers and \(a \ne 0\).

Q: Name the methods of solving a quadratic equation.

By factorisation, by completing the square, and by using the quadratic formula.

Q: Write the quadratic formula.

For \(ax^2 + bx + c = 0\), \(x = \dfrac{-b \pm \sqrt{b^2 - 4ac}}{2a}\).

Q: What is the discriminant, and what does it tell us?

The discriminant is \(D = b^2 - 4ac\); if \(D > 0\) the roots are real and distinct, if \(D = 0\) real and equal, and if \(D < 0\) there are no real roots.

Q: Solve \(x^2 - 5x + 6 = 0\) by factorisation.

\((x - 2)(x - 3) = 0\), so \(x = 2\) or \(x = 3\).

Q: Find the nature of the roots of \(2x^2 - 3x + 5 = 0\).

\(D = (-3)^2 - 4(2)(5) = 9 - 40 = -31 < 0\), so the equation has no real roots.

Q: For what value of \(k\) does \(x^2 + kx + 4 = 0\) have equal roots?

Equal roots need \(D = 0\): \(k^2 - 16 = 0\), so \(k = \pm 4\).

Q: The product of two consecutive positive integers is 56. Frame and solve the equation.

Let the integers be \(x\) and \(x + 1\): \(x(x + 1) = 56 \Rightarrow x^2 + x - 56 = 0 \Rightarrow (x + 8)(x - 7) = 0\), so \(x = 7\) and the integers are 7 and 8.

Try one from this chapter

The standard form of a quadratic equation is:

Summary

A quadratic equation has the standard form \(ax^2 + bx + c = 0\) with \(a \ne 0\). This chapter teaches how to solve such equations by factorisation and by the quadratic formula \(x = \dfrac{-b \pm \sqrt{b^2 - 4ac}}{2a}\). The expression \(b^2 - 4ac\), called the discriminant, decides the nature of the roots: two distinct real roots if it is positive, two equal real roots if it is zero, and no real roots if it is negative. Many practical problems about areas, speeds and ages reduce to quadratic equations, so mastering these methods is essential.

Standard formSolving by factorisationQuadratic formulaNature of roots (discriminant)Applications

Key terms

Quadratic equation

An equation of the form \(ax^2 + bx + c = 0,\ a \ne 0\).

Discriminant

The quantity \(b^2 - 4ac\) that determines the nature of the roots.

Roots

The values of \(x\) that satisfy the quadratic equation.

Quadratic formula

\(x = \dfrac{-b \pm \sqrt{b^2 - 4ac}}{2a}\).

Extra questions & answers

EXERCISE 4.1

EXERCISE 4.2

EXERCISE 4.3

Explore interactively

Explore the parabola, roots and discriminant

abc

Discriminant

Roots

2, 1

Vertex

(1.5, -0.25)

Discriminant > 0 — two distinct real roots.

Key-term flashcards

Flip cards · mark known · keyboard ← → and Space

4 cards

Term1 / 4

Quadratic equation

Tap to reveal

Meaning1 / 4

An equation of the form \(ax^2 + bx + c = 0,\ a \ne 0\).

Tap to flip back

Tap card to flip

Keyboard: ← → to move · Space to flip

Practice quiz · Quadratic Equations

Score on this chapter, climb the leaderboard, and get an AI diagnosis of your mistakes.

Dual AI-verified questions Real exam pattern 1 free quiz, first unlock ₹1, then ₹9

Quadratic Equations

Start here

Maths 10 Qs · ~10 min

Instant score, answer review and weak-topic diagnosis

7.5/10avg score

9/10best

Your Class 10 board-year rhythm

1
A little every day
Chapter notes and one quiz a day beat a month of last-minute cramming.
2
Board-pattern practice
Extra questions with answers and quizzes asked the way the board asks them.
3
Results you can show off
Scorecards, leaderboards and AI analysis of every attempt.