Quadratic Equation Class 10 ||Maths|| Chapter 4 NCERT Notes

Quadratic Equation Class 10 ||Maths|| Chapter 4 NCERT Notes

 A quadratic equation is a polynomial equation of degree 2. The standard form of a quadratic equation is:

ax2+bx+c=0

where:

  • a≠0,
  • b and c are constants,
  • x is the variable.

1. What is a Quadratic Equation?

A quadratic equation is an equation of the form:

ax2+bx+c=0wherea≠0

  • Examples: 2x2+3x+1=0, x2−4=0
  • Not a quadratic equation: x+5=0 (this is linear).

2. Roots of a Quadratic Equation

The solutions of the quadratic equation are called its roots or zeros. These are the values of x that satisfy the equation. A quadratic equation can have:

  • Two real and distinct roots,
  • Two real and equal roots,
  • No real roots (when the roots are imaginary).

3. Methods to Solve a Quadratic Equation

There are three main methods to solve quadratic equations:

i. Factorization Method

  1. Factorization involves breaking down the middle term (the term with x) into two parts that multiply to give the product of a×c.

  2. Example:

    2x2+7x+3=0

    Step 1: Multiply a and c: 2×3=6.

    Step 2: Find two numbers that multiply to give 6 and add up to 7: these are 6 and 1.

    Step 3: Rewrite the middle term:

    2x2+6x+x+3=0

    Step 4: Factorize:

    2x(x+3)+1(x+3)=0

    Step 5: Combine factors:

    (2x+1)(x+3)=0

    Step 6: Set each factor to 0:

    2x+1=0orx+3=0

    The roots are x=−12 and x=−3.

ii. Completing the Square Method

This method involves converting the quadratic equation into a perfect square trinomial. The steps are:

  1. Start with the quadratic equation:ax2+bx+c=0
  2. Divide the whole equation by a (if a≠1):x2+bax+ca=0
  3. Move the constant term to the other side:x2+bax=−ca
  4. Add (b2a)2 to both sides to complete the square:x2+bax+(b2a)2=−ca+(b2a)2
  5. Express the left side as a square:(x+b2a)2=some value
  6. Take the square root of both sides and solve for x.

iii. Quadratic Formula

The quadratic formula is a universal method to solve any quadratic equation. For a quadratic equation ax2+bx+c=0, the roots are given by:

x=−b±b2−4ac2a

Where:

  • b2−4ac is called the discriminant (D).
    • If D>0: The equation has two real and distinct roots.
    • If D=0: The equation has two real and equal roots.
    • If D<0: The equation has no real roots (the roots are imaginary).

Example: Solve 2x2−4x−6=0:

  1. Identify coefficients: a=2, b=−4, c=−6.
  2. Apply the quadratic formula:x=−(−4)±(−4)2−4(2)(−6)2(2)
  3. Simplify:x=4±16+484=4±644=4±84
  4. Solve for x:x=4+84=3orx=4−84=−1

The roots are x=3 and x=−1.

4. Nature of Roots

The discriminant (D) helps determine the nature of the roots:

D=b2−4ac

  • D > 0: Two real and distinct roots.
  • D = 0: Two real and equal roots.
  • D < 0: No real roots (imaginary roots).

5. Word Problems Involving Quadratic Equations

Many practical problems lead to quadratic equations, especially those involving areas, motion, and finance. Follow these steps for solving word problems:

  1. Identify the unknown quantities.
  2. Frame the quadratic equation based on the given information.
  3. Solve the quadratic equation using one of the methods discussed.
  4. Interpret the solution in the context of the problem.

Example: The area of a rectangular plot is 528 m². The length of the plot is 2 m more than twice its breadth. Find the length and breadth.

  1. Let the breadth be x meters. Then, the length is 2x+2 meters.
  2. The area is 528 m², so:x(2x+2)=5282x2+2x−528=0
  3. Solve this quadratic equation to find x.

Key Concepts:

  • Quadratic Equation: ax2+bx+c=0
  • Discriminant: D=b2−4ac (determines the nature of roots)
  • Methods of solving: Factorization, Completing the square, Quadratic formula
  • Nature of Roots: Based on the discriminant