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Algebra

Algebra: Learn Step by Step

Simple explanations, formulas, solved examples, and practice questions for students.
Algebra Home Tutor

Simple & Easy Algebra

Algebra helps us solve unknown values using symbols and equations. It is one of the most important topics in Mathematics.

Algebra Learning

1. What is Algebra?

Algebra is a branch of mathematics where letters and symbols are used to represent numbers. These letters are called variables.

Example:
x + 5 = 10
Here, x is an unknown number.

To find x:
x = 10 - 5
x = 5
Practice:
1. Find x if x + 7 = 20
2. Find y if y - 4 = 12
3. Find z if z + 15 = 40

2. Variables and Constants

A variable is a symbol whose value can change. A constant is a fixed value.

In the expression:
3x + 5
x → Variable
3 and 5 → Constants

Types of Variables

  • x, y, z are common variables
  • Variables may represent age, distance, marks etc.
  • They can take different values
Identify variables and constants:
1. 7a + 9
2. 10x - 4
3. 2y + 12

3. Algebraic Expressions

An algebraic expression contains variables, constants, and mathematical operations.

2x + 3
This is an algebraic expression.

Types of Expressions

  • Monomial → One term → 5x
  • Binomial → Two terms → x + 5
  • Trinomial → Three terms → x2 + 2x + 1
Simplification Example:
3x + 2x = 5x
Like terms can be added.
Simplify:
1. 5x + 3x
2. 10y - 4y
3. 7a + 2a - a

4. Algebraic Equations

An equation states that two expressions are equal.

2x + 3 = 11
Step 1: Subtract 3 from both sides
2x = 8
Step 2: Divide both sides by 2
x = 4
Another Example:
5y - 10 = 20
Add 10 both sides:
5y = 30
Divide by 5:
y = 6
Solve:
1. x + 8 = 15
2. 3x = 21
3. 4y - 8 = 12

5. Algebraic Identities

Algebraic identities are formulas that are always true.

Identity 1:
(a + b)² = a2 + 2ab + b2
Example:
(x + 2)² = x2 + 4x + 4
Identity 2:
(a - b)² = a2 - 2ab + b2
Identity 3:
a2 - b2 = (a + b)(a - b)
Expand:
1. (x + 3)²
2. (a - 5)²
3. (y + 2)(y - 2)

6. Factorization

Factorization means writing an expression as multiplication of factors.

x2 + 5x = x(x + 5)
x2 - 9 = (x + 3)(x - 3)
Factorize:
1. x2 + 4x
2. y2 - 16
3. 3a + 6

7. Linear Equations in One Variable

A linear equation has variable power 1.

2x + 5 = 17
Step 1:
2x = 12
Step 2:
x = 6
Word Problem:

Rahul has some chocolates. After getting 5 more, he has 20 chocolates. How many did he have earlier?
x + 5 = 20
x = 15
Practice:
1. 5x + 10 = 35
2. 8y - 16 = 24
3. 3a + 7 = 19

8. Quick Algebra Quiz

Question: Solve: 2x + 6 = 18

8. Quadratic Equation

A quadratic equation is an equation where the highest power of the variable is 2. These equations are very important in algebra and are used in physics, engineering, graphs, and daily mathematical calculations.

ax2 + bx + c = 0
where a ≠ 0

Here,
a = coefficient of x2
b = coefficient of x
c = constant term

Example:
x2 + 5x + 6 = 0
Reference:
Quadratic equations usually form parabola graphs.

9. Roots of Quadratic Equation

Roots are the values of x that satisfy the quadratic equation. In simple words, roots make the equation equal to zero.

x = -b ± √(b2 - 4ac)2a
This is called the quadratic formula.
Example:
x2 - 5x + 6 = 0
Roots are 2 and 3.

9. Roots of Quadratic Equation

x = -b ± √(b2 - 4ac)2a
This is called the quadratic formula.

10. Sum and Product of Roots

If α and β are roots of a quadratic equation, then we can easily find their sum and product using formulas.

α + β = -ba
αβ = ca
Example:
x2 - 7x + 10 = 0
Sum of roots = 7
Product of roots = 10

11. Condition for Real and Equal Roots

The nature of roots depends on the value of the discriminant.

D = b2 - 4ac
  • If D > 0 → roots are real and different
  • If D = 0 → roots are real and equal
  • If D < 0 → roots are imaginary
b2 - 4ac = 0
If discriminant is zero, roots are real and equal.

11. Condition for Real and Equal Roots

b2 - 4ac = 0
If discriminant is zero, roots are real and equal.

12. Arithmetic Progression (AP)

Arithmetic Progression is a sequence where the difference between consecutive terms is constant.

2, 5, 8, 11, 14...
Common difference = 3
nth term = a + (n-1)d

Here,
a = first term
d = common difference

13. Geometric Progression (GP)

Geometric Progression is a sequence where every term is multiplied by a fixed number.

2, 4, 8, 16...
Common ratio = 2
nth term = ar^(n-1)

GP is used in compound interest and population growth problems.

13. Geometric Progression (GP)

a, ar, ar2...
Common ratio = r
nth term = ar^(n-1)

14. Binomial Expansion

Binomial expansion is used to expand powers of expressions like (a+b)n.

(a+b)2 = a2 + 2ab + b2
(x+3)2 = x2 + 6x + 9

Binomial theorem helps to expand large powers quickly.

15. Combination Formula

Combination is used when arrangement order does not matter.

nCr = n!r!(n-r)!
Example: Choosing 2 students from 5 students.

15. Combination Formula

nCr = n!r!(n-r)!

16. Permutation Formula

Permutation is used when arrangement order matters.

nPr = n!(n-r)!
Example: Arranging 3 students in a line from 5 students.

17. Factorial Definition

Factorial means multiplication of all natural numbers from the number to 1.

n! = n × (n-1) × (n-2)...1
Example: 5! = 120
4! = 4 × 3 × 2 × 1 = 24

17. Factorial Definition

n! = n × (n-1) × (n-2)...1
Example: 5! = 120

18. Logarithmic Identities

Logarithmic identities help simplify difficult logarithm expressions.

log(ab) = log a + log b
log(a/b) = log a - log b
log(an) = n log a
Example:
log(100) = 2
because 102 = 100

19. Change of Base Formula

This formula helps to convert logarithms from one base to another.

loga b = logc blogc a
This is useful in calculators where only common logarithms are available.

19. Change of Base Formula

loga b = logc blogc a

20. Exponential and Logarithmic Relation

Exponential and logarithmic forms are inverse of each other.

ax = y ⇔ log_a y = x
Example:
23 = 8
therefore,
log2 8 = 3

21. Arithmetic Mean

Arithmetic Mean is the average of numbers.

AM = a+b2
Example: Average of 4 and 8:
(4+8)/2 = 6

21. Arithmetic Mean

AM = a+b2

22. Geometric Mean

Geometric Mean is found by taking square root of the product of numbers.

GM = √ab
Example:
GM of 4 and 9 = √36 = 6

23. Harmonic Mean

Harmonic Mean is used in average speed and rate problems.

HM = 2aba+b
Example:
HM of 4 and 12 = 6

23. Harmonic Mean

HM = 2aba+b

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