Mathematics Notes for Class 10 Quadratic Equations

Quadratic Equations

Welcome to the chapter on Quadratic Equations for Class 10. In this chapter, you will learn what quadratic equations are, how to solve them, and how they are used in real-life situations. By the end of this chapter, you will be able to recognize, form, and solve quadratic equations using different methods.

Key Concepts

Quadratic Equation: An equation of the form ax² + bx + c = 0, where a, b, and c are real numbers and a ≠ 0.
Roots/Solutions: The values of x that satisfy the quadratic equation.
Discriminant (D): D = b² - 4ac, used to determine the nature of the roots.

Methods to Solve Quadratic Equations

Factorization Method: Express the equation as a product of two linear factors and set each factor to zero.
Completing the Square: Rearrange the equation to make one side a perfect square trinomial.
Quadratic Formula: x = [-b ± √(b² - 4ac)] / (2a)

Nature of Roots

If D > 0: Two distinct real roots.
If D = 0: Two equal real roots.
If D < 0: No real roots (roots are complex).

Forming Quadratic Equations

Quadratic equations can be formed from word problems or given roots. If α and β are roots, the equation is x² - (α + β)x + αβ = 0.

Applications of Quadratic Equations

Solving problems related to area, speed, age, and geometry.
Used in physics, engineering, and many real-life situations.

Practice Questions

Solve: x² - 5x + 6 = 0 by factorization.
Find the roots of 2x² + 3x - 2 = 0 using the quadratic formula.
What is the nature of the roots of x² + 4x + 5 = 0?
Form a quadratic equation whose roots are 3 and -2.
A rectangular garden has an area of 60 m². Its length is (x + 5) m and width is (x - 3) m. Find x.

Challenge Yourself

Prove that the sum and product of the roots of ax² + bx + c = 0 are -b/a and c/a respectively.
If one root of the equation x² + px + 12 = 0 is 3, find the value of p and the other root.

Did You Know?

Quadratic equations were studied as early as 2000 BC in Babylonian mathematics!
The graph of a quadratic equation is always a parabola.

Glossary

Quadratic Equation: An equation of the form ax² + bx + c = 0.
Root: A solution of the equation.
Discriminant: D = b² - 4ac, helps determine the nature of roots.
Parabola: The U-shaped curve formed by the graph of a quadratic equation.

Answers to Practice Questions

x² - 5x + 6 = 0 ⇒ (x - 2)(x - 3) = 0 ⇒ x = 2, 3
x = [-3 ± √(9 + 16)] / 4 = [-3 ± 5]/4 ⇒ x = 0.5, x = -2
D = 16 - 20 = -4 < 0, so roots are complex (no real roots).
x² - (3 + (-2))x + (3 × -2) = x² - x - 6 = 0
(x + 5)(x - 3) = 60 ⇒ x² + 2x - 15 = 60 ⇒ x² + 2x - 75 = 0

Practice solving quadratic equations to master algebra and solve real-world problems!

Mathematics Class 10 - Quadratic Equations Notes