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Business Mathematics and Basic Statistics · Class 11 Commerce

Ch 1Quadratic Equations — Class 11 Business Mathematics and Basic Statistics, concept-first.

A quadratic equation in one variable is an equation of degree 2, written in its general form as

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Key concepts

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Solving by Factorization

Rewrite as a product of two linear factors by splitting the middle term into two parts whose product is and whose sum is , then apply the zero-product rule: a product is zero only when one of its factors is zero.

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Chapter contents

The NCERT structure, section by section. Open a section to see its questions, then read the concept-first solution.

1

General Form of a Quadratic Equation

A quadratic equation in one variable is an equation of degree 2, written in its general form as

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Zeros (Roots) of a Quadratic Polynomial

A root (also called a zero) of the quadratic equation is a real number such that substituting into makes the expression equal to zero, i.e. .

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Solving by Factorization

Factorization solves by rewriting the left side as a product of two linear factors, then using the fact that a product of real numbers is zero only when at least one factor is zero (the zero-product r…

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Solving by the Method of Perfect Squares

The method of perfect squares (also called "completing the square") solves by rewriting the quadratic expression so that the -terms form a perfect-square trinomial, , and then taking a square root on…

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Sridharacharya's Method (Statement)

Sridharacharya's rule gives the roots of () directly, as a formula, without repeating the completing-the-square steps each time:

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The Discriminant and the Number of Real Roots

The quantity appearing under the square root in Sridharacharya's formula is called the discriminant of the quadratic equation.

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Verifying Whether a Given Number is a Root

Sometimes a problem gives a candidate value and asks only whether it is a root of a given quadratic equation, without asking to solve the equation from scratch.

Exercises

+Show 4 questions4 questions
  1. Q10Solve by factorization: $x^2 + 7x + 12 = 0$.Free
  2. Q11Solve by the method of perfect squares: $x^2 + 2x - 5 = 0$.Free
  3. Q12Using Sridharacharya's formula, solve $4x^2 - 4x + 1 = 0$ and comment on the nature of the roots.Preview
  4. Q13Without solving, determine the number of real roots of $x^2 + 2x + 5 = 0$ using the discriminant, and also verify whether $x = -1$ is a root…Preview

More questions

+Show 9 questions9 questions
  1. Example 1Solve the quadratic equation $x^2 - 5x + 6 = 0$ by factorization.Free
  2. Example 2Solve $2x^2 + 3x - 2 = 0$ by factorization.Free
  3. Example 3Solve $x^2 - 6x + 7 = 0$ by the method of perfect squares (completing the square).Free
  4. Example 4Solve $2x^2 - 4x - 3 = 0$ by the method of perfect squares.Preview
  5. Example 5Using Sridharacharya's formula, solve $3x^2 - 5x - 2 = 0$.Preview
  6. Example 6Using Sridharacharya's formula, solve $x^2 - 4x + 4 = 0$ and state what this tells us about the number of real roots.Preview
  7. Example 7Without solving the equation, find the number of real roots of $2x^2 + 3x + 5 = 0$ using the discriminant.Preview
  8. Example 8Verify whether $x = 3$ is a root of the equation $x^2 - x - 6 = 0$.Preview
  9. Example 9Verify whether $x = -1$ is a root of the equation $2x^2 + 5x + 3 = 0$. If it is, find the other root using the sum-of-roots relation.Preview