Quadratic Equations

Board: GSEB | Class: Std 10

Comprehensive study notes for Quadratic Equations by Ajay Yadav (Math King of Katargam). Master every concept with clear explanations, solved examples, and practice problems.

Key Concepts

Standard Form

A quadratic equation in x is of the form ax² + bx + c = 0 where a ≠ 0. The highest power of the variable is 2.

Solving by Factorization

Write ax² + bx + c as (px + q)(rx + s) = 0. Then set each factor to zero: px + q = 0 or rx + s = 0. This gives the roots.

Solving by Completing the Square

Rewrite ax² + bx + c = 0 as (x + b/2a)² = (b² – 4ac)/4a². Take square root on both sides. This method derives the quadratic formula.

Quadratic Formula

x = (-b ± √b² – 4ac)/2a. The expression D = b² – 4ac is called the discriminant.

Nature of Roots

D > 0: Two distinct real roots. D = 0: One real root (repeated). D < 0: No real roots (complex conjugates).

Word Problems

Quadratic equations model many real-world situations: area problems, number problems, time-speed-distance, age problems, and profit-loss scenarios.

Important Formulas

Standard Form	`ax² + bx + c = 0, a ≠ 0`
Quadratic Formula	`x = (-b ± √D)/2a where D = b² - 4ac`
Discriminant	`D = b² - 4ac`
D > 0	`Two distinct real roots`
D = 0	`One real root (repeated)`
D < 0	`No real roots`

Solved Examples

Example 1: Solve x² – 5x + 6 = 0 by factorization.

Solution: x² – 5x + 6 = (x-2)(x-3) = 0. x = 2 or x = 3.

Example 2: Solve 2x² – 5x + 3 = 0 using the formula.

Solution: a=2, b=-5, c=3. D = 25 – 24 = 1. x = (5 ± 1)/4. x = 3/2 or x = 1.

Example 3: Find the discriminant of x² + 4x + 5 = 0.

Solution: D = 16 – 20 = -4. Since D < 0, no real roots exist.

Practice Questions

Solve: 2x² – 7x + 3 = 0.
Find k such that x² + kx + 4 = 0 has equal roots.
The area of a rectangle is 240 cm². Its length is 8 cm more than width. Find dimensions.
Solve: 9x² + 7x – 2 = 0 using the quadratic formula.
If the sum of two numbers is 15 and sum of squares is 113, find the numbers.

Download PDF

Click here to download the PDF notes for this chapter.

Video Lessons

Watch video explanations on our Videos page.