‘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.

Quadratic Equations

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.

Class 10 Mathematics · Chapter 4

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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

Try one from this chapter

The standard form of a quadratic equation is:

Flashcards & interactive practice

Quadratic Equations

CBSE Class 10 Board Examination community

Summary

Key terms

Extra questions & answers

EXERCISE 4.1

EXERCISE 4.2

EXERCISE 4.3

Practice quiz · Quadratic Equations

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