Class 10 Notes - Quadratic Equations

Quadratic Equations

A quadratic equation is a second-degree polynomial equation in one variable, usually written as:
ax² + bx + c = 0, where a ≠ 0.

Standard Form

a, b, c are real numbers
a ≠ 0 (if a = 0, the equation becomes linear)

Examples of Quadratic Equations

2x² + 3x - 5 = 0
x² - 4x + 4 = 0
3x² + 7 = 0

Methods to Solve Quadratic Equations

Factorization Method
- Express the quadratic equation in the form ax² + bx + c = 0
- Factorize the quadratic expression and set each factor to zero
- Solve for x
Completing the Square
- Rewrite the equation so that one side is a perfect square trinomial
- Take the square root of both sides and solve for x
Quadratic Formula
- The roots of ax² + bx + c = 0 are given by:
  x = [-b ± √(b² - 4ac)] / (2a)

Nature of Roots

Discriminant D = b² - 4ac
If D > 0: Two distinct real roots
If D = 0: Two equal real roots
If D < 0: Two complex roots (no real roots)

Word Problem Example

Problem: The product of two consecutive positive integers is 56. Find the integers.

Solution:

Let the integers be x and x+1.
x(x+1) = 56 ⇒ x² + x - 56 = 0
Factorizing: (x + 8)(x - 7) = 0 ⇒ x = -8 or x = 7
Since integers are positive, x = 7. So, the integers are 7 and 8.

Practice Questions

Solve: x² - 5x + 6 = 0
Solve using the quadratic formula: 2x² + 3x - 2 = 0
Find the nature of roots for: x² + 4x + 5 = 0
If the sum and product of the roots of a quadratic equation are 5 and 6 respectively, write the equation.

Summary

Quadratic equations are of the form ax² + bx + c = 0, a ≠ 0
They can be solved by factorization, completing the square, or using the quadratic formula
The nature of roots depends on the discriminant (D = b² - 4ac)
Quadratic equations are widely used in real-life problem solving