Unit 7: Quadratic Equation (वर्ग समीकरण) Formulae & Notes
Introduction
Unit 7: Quadratic Equation (वर्ग समीकरण) is a core part of Algebra. This unit covers solving simultaneous linear equations through substitution and elimination, as well as mastering quadratic equations using factorization and the quadratic formula. Understanding the nature of roots and algebraic identities is essential for this chapter.
1. Important Algebraic Formulas
These algebraic identities are fundamental for factorizing and solving both linear and quadratic equations.
| Identity Name | Formula |
|---|---|
| Difference of Two Squares |
$$a^2 – b^2 = (a + b)(a – b)$$
Useful for simplifying expressions like $aw+bw = a^2-b^2$ |
| Square of a Sum | $$(a + b)^2 = a^2 + 2ab + b^2$$ |
| Square of a Difference | $$(a – b)^2 = a^2 – 2ab + b^2$$ |
2. Simultaneous Linear Equations
Key Concepts
- Definition: A set of two or more equations with the same variables (usually $x$ and $y$) solved together.
- Substitution Method: Solve one equation for a variable (e.g., $y = \dots$) and substitute into the other.
- Elimination Method: Add or subtract equations to eliminate one variable.
Word Problems (Verbal Problems) Translation
| Verbal Statement | Algebraic Translation |
|---|---|
| Sum of two numbers is 20 | $$x + y = 20$$ |
| A number of two digits | $$10x + y$$(x = tens digit, y = units digit) |
| If 1 is added to numerator… | $$\frac{x+1}{y}$$ |
| Five years ago… (Age) | $$(x – 5) \text{ and } (y – 5)$$ |
Tip: Always identify two conditions in the problem. Each condition creates one equation.
3. Quadratic Equations
Standard Form
$$ax^2 + bx + c = 0$$
Where $a \neq 0$
Quadratic Formula
$$x = \frac{-b \pm \sqrt{b^2 – 4ac}}{2a}$$
Nature of Roots (Discriminant $D = b^2 – 4ac$)
[Image of quadratic discriminant graph]| Value of D | Nature of Roots |
|---|---|
| $$D > 0$$ | Roots are real and unequal. |
| $$D = 0$$ | Roots are real and equal. |
| $$D < 0$$ | Roots are imaginary (no real solution). |
4. Analysis of Specific Exercises
Type A: Variable Isolation
Example: If $aw + bw = a^2 – b^2$, find $w$.
Step 1 (Factorize LHS): w(a + b)
Step 2 (Factorize RHS): (a + b)(a – b)
Step 3 (Solve): w = (a+b)(a-b) / (a+b)
Result: $$w = a – b$$
Step 2 (Factorize RHS): (a + b)(a – b)
Step 3 (Solve): w = (a+b)(a-b) / (a+b)
Result: $$w = a – b$$
Type B: Substitution Values
Example: If $y = \frac{4x + 4}{3}$ and $x=2$, find $y$.
$$y = \frac{4(2) + 4}{3} = \frac{8 + 4}{3} = \frac{12}{3} = 4$$
Full Chapter PDF Manual
For offline study, access the complete PDF manual below.
Disclaimer: This content is based on the Class 10 Compulsory Mathematics manual by Dr. Simkhada (Readmore Publishers).