Algebra: Complete Concepts, Formulas, Shortcuts and Solved Examples
Master algebra step by step with clear concepts, essential formulas, smart shortcuts, worked examples, and practice questions.
This complete guide introduces algebra from the ground up and builds towards the types of equations and expressions used in school mathematics, aptitude tests, placement assessments, and competitive examinations such as SSC, RRB and banking exams.
Learn how symbols represent numbers, how to simplify expressions, how to solve equations, and how to choose an efficient method without memorising tricks blindly.
1. What Is Algebra?
Algebra is a branch of mathematics that uses letters and symbols to represent numbers and relationships. Instead of working only with known numbers, algebra lets us describe a general rule or find an unknown value.
Known numbers
All values are already given.
An unknown value
Find the value of the variable.
Many possible values
The rule works for any suitable values of a and b.
Why is algebra useful?
Algebra helps you solve unknowns, model real-world situations, recognise patterns, and calculate efficiently. It supports later topics such as equations, graphs, geometry, percentages, ratios, and quantitative aptitude.
A number increased by 8 is 21. Find the number.
2. Basic Concepts and Vocabulary
Variable
A letter that represents a value that may be unknown or may change.
Example: In x + 4 = 10, x is the variable.
Constant
A fixed number that does not change within the expression.
Example: In 3x + 7, the constant term is 7.
Coefficient
The numerical factor multiplying a variable.
In −2y, the coefficient is −2.
Term
A part of an expression separated by addition or subtraction signs.
The terms are 4x, 3y, and −8.
Expression, equation, and formula
| Term | Meaning | Example |
|---|---|---|
| Expression | A mathematical phrase without an equals sign. | 3x + 7 |
| Equation | A statement that two expressions are equal. | 3x + 7 = 22 |
| Formula | A rule connecting quantities. | A = l × w |
| Identity | An equality true for every value in its domain. | (a + b)² = a² + 2ab + b² |
Like and unlike terms
Like terms have the same variable part with the same powers. Their coefficients can be combined. Unlike terms cannot be combined into one term.
Simplify: 3x + 5 − 2x + 7.
Combine the x-terms and the constants separately:
But 3x + 2y cannot be simplified to 5xy or 5x. The variables are different.
3. Fundamental Rules of Algebra
Rule 1 · Commutative property
The order can be changed for addition and multiplication.
Subtraction and division are not generally commutative.
Rule 2 · Associative property
Regrouping does not change a sum or a product.
Rule 3 · Distributive property
Multiply the factor outside the brackets by every term inside.
Simplify 4(x + 3).
Rule 4 · Equality stays balanced
You may add, subtract, multiply, or divide both sides of an equation by the same appropriate value. Never divide by zero.
When multiplying or dividing an inequality by a negative number, reverse the inequality sign.
Rule 5 · Sign rules
| Operation | Result |
|---|---|
| (+) × (+) | Positive |
| (−) × (−) | Positive |
| (+) × (−) | Negative |
| (−) × (+) | Negative |
The same sign pattern applies to division when the divisor is non-zero.
4. Simplifying Algebraic Expressions
To simplify an expression, remove brackets when appropriate, apply sign rules, and combine like terms. Follow the order of operations: brackets first, powers, multiplication and division, then addition and subtraction.
Simplify: 3(2x − 4) + 5x − 2.
Removing a minus sign before brackets
A minus sign before a bracket changes the sign of every term inside.
Simplify: 8x − (3x − 5).
Change both signs inside the bracket:
5. Important Algebraic Identities
Algebraic identities are useful for expanding expressions and recognising patterns quickly. Learn the patterns, but always check that the signs and middle terms are correct.
Square of a sum
Example: (x + 3)² = x² + 6x + 9.
Square of a difference
Example: (x − 4)² = x² − 8x + 16.
Difference of squares
Example: x² − 25 = (x − 5)(x + 5).
Product of two binomials
Example: (x + 2)(x + 5) = x² + 7x + 10.
Cube of a sum
Cube of a difference
Sum of cubes
Difference of cubes
Calculate 49² using an identity.
Write 49 as 50 − 1.
6. Factorisation
Factorisation means rewriting an expression as a product of simpler expressions. It reverses expansion and is especially useful for simplifying algebraic fractions and solving equations.
Method 1 · Take out a common factor
Factorise 6x + 12.
The highest common factor is 6.
Method 2 · Factorise a quadratic trinomial
For x² + bx + c, look for two numbers that add to b and multiply to c.
Factorise x² + 7x + 12.
Find two numbers whose sum is 7 and product is 12: 3 and 4.
Method 3 · Difference of two squares
Factorise 9x² − 16.
Recognise the squares: (3x)² − 4².
7. Solving Linear Equations
A linear equation in one variable has the variable to the first power. The goal is to isolate the variable while keeping both sides balanced.
Solve: 3x + 7 = 25.
Solve: 5x − 4 = 2x + 11.
Solve: 2(3x − 1) = 16.
Expand, then isolate x:
8. Fractions, Powers and Quadratic Equations
Algebraic fractions
To solve equations containing fractions, multiply every term on both sides by the least common denominator. Check that you do not multiply only some terms.
Solve: x/3 + 2 = 5.
Laws of exponents
| Rule | Formula | Example |
|---|---|---|
| Multiply same base | aᵐ × aⁿ = aᵐ⁺ⁿ | x² × x³ = x⁵ |
| Divide same base | aᵐ ÷ aⁿ = aᵐ⁻ⁿ | x⁵ ÷ x² = x³, for x ≠ 0 |
| Power of a power | (aᵐ)ⁿ = aᵐⁿ | (x²)³ = x⁶ |
| Power of a product | (ab)ⁿ = aⁿbⁿ | (2x)² = 4x² |
| Zero exponent | a⁰ = 1, for a ≠ 0 | 5⁰ = 1 |
| Negative exponent | a⁻ⁿ = 1/aⁿ, for a ≠ 0 | x⁻² = 1/x² |
Quadratic equations: a first look
A quadratic equation has the form ax² + bx + c = 0, where a is not zero. Depending on the coefficients, it may be solved by factorisation, completing the square, or the quadratic formula.
The expression b² − 4ac is called the discriminant. For real coefficients, it helps determine the type of roots.
Solve: x² − 5x + 6 = 0.
Find two numbers that multiply to 6 and add to −5: −2 and −3.
If a product is zero, at least one factor must be zero.
9. Algebra Tricks and Shortcuts
Shortcuts work best when you recognise the pattern. Use them to reduce steps, not to skip the logic.
Combine like terms first
In 8x + 3 − 2x + 7, combine x-terms and constants:
This keeps the expression tidy before further operations.
Use identities near round numbers
For 99², write 99 = 100 − 1.
Move terms carefully
Instead of saying “move it and change the sign,” think of doing the same operation to both sides.
Factor before multiplying
Recognise x² − 49 as a difference of squares.
Check by substitution
If you find x = 4, substitute 4 into the original equation. This catches sign and arithmetic errors quickly.
Clear fractions early
For an equation with denominators 2 and 3, multiply every term by 6 to remove the fractions, while keeping the equation balanced.
Choose a variable for the unknown
For “three more than twice a number,” write 2x + 3—not 3x + 2.
Look for a common factor
In 12x + 18, take out 6.
10. Turning Word Problems into Equations
Algebra becomes powerful when words are converted into mathematical statements. Define the unknown first, then translate each phrase carefully.
| Words | Algebraic form |
|---|---|
| A number increased by 7 | x + 7 |
| A number decreased by 4 | x − 4 |
| Twice a number | 2x |
| Three times a number, then add 5 | 3x + 5 |
| Five less than twice a number | 2x − 5 |
| One number is 8 more than another | x = y + 8 |
| The sum of two consecutive integers | x + (x + 1) |
The sum of a number and twice that number is 36. Find the number.
The sum of two consecutive integers is 41. Find the integers.
Let the first integer be x. The next integer is x + 1.
11. Common Algebra Mistakes
Combining unlike terms
Incorrect: 3x + 2y = 5xy.
Correct: 3x + 2y stays as it is because x and y are unlike variables.
Squaring a sum incorrectly
Incorrect: (a + b)² = a² + b².
Correct: (a + b)² = a² + 2ab + b².
Forgetting to distribute
Incorrect: 3(x + 4) = 3x + 4.
Correct: 3(x + 4) = 3x + 12.
Sign error with brackets
Incorrect: x − (y − 2) = x − y − 2.
Correct: x − (y − 2) = x − y + 2.
Dividing by zero
Division by zero is undefined. Check denominators before simplifying algebraic fractions.
Not checking the answer
Substitute the solution into the original equation. If both sides are equal, the solution passes the check.
12. How to Prepare for Algebra
Build confidence in layers. Do not rush to advanced identities before you can simplify expressions and solve one-step equations accurately.
| Day | Focus | Practice target |
|---|---|---|
| 1 | Variables, terms and expressions | 15 questions |
| 2 | Like terms, brackets and sign rules | 15 questions |
| 3 | Identities and expansion | 12 questions |
| 4 | Factorisation | 12 questions |
| 5 | Linear equations and fractions | 15 questions |
| 6 | Word problems and exponents | 15 questions |
| 7 | Mixed revision and a timed quiz | 20 questions |
A practical study routine
- Read the concept and copy the formula in your own words.
- Study one worked example, then cover the solution and redo it.
- Attempt 10–15 questions without looking at the answer key.
- Record mistakes by type: concept, sign, equation setup, or arithmetic.
- Reattempt incorrect questions the next day.
13. Practice Questions
Try each question before opening the solution. The questions move from basic operations to equations, identities and factorisation.
Q1 · Basic Simplify: 7x + 3x − 4.
Solution: Combine like terms: 7x + 3x = 10x.
Trick: Combine variable terms separately from constants.
Q2 · Basic If x + 9 = 23, find x.
Subtract 9 from both sides: x = 23 − 9.
Trick: Undo addition with subtraction.
Q3 · Basic Solve: 5x = 45.
Divide both sides by 5.
Trick: Undo multiplication with division.
Q4 · Intermediate Simplify: 3(2x + 5) − 4x.
Expand: 6x + 15 − 4x. Combine like terms.
Trick: Expand brackets before combining terms.
Q5 · Intermediate Solve: 4x − 7 = 21.
Add 7: 4x = 28. Divide by 4.
Q6 · Intermediate Solve: 3x + 4 = x + 18.
Subtract x: 2x + 4 = 18. Subtract 4: 2x = 14. Therefore x = 7.
Q7 · Identity Expand: (x + 5)².
Use (a + b)² = a² + 2ab + b².
Trick: The middle term is twice the product of the two parts.
Q8 · Identity Factorise: x² − 36.
This is a difference of squares: x² − 6².
Q9 · Factorisation Factorise: x² + 9x + 20.
Find two numbers that add to 9 and multiply to 20: 4 and 5.
Q10 · Intermediate Solve: 2(x − 3) = 18.
Divide both sides by 2: x − 3 = 9. Add 3.
Q11 · Fractions Solve: x/4 + 3 = 8.
Subtract 3: x/4 = 5. Multiply by 4.
Q12 · Exponents Simplify: x³ × x⁴.
For the same base, add the exponents: 3 + 4 = 7.
Q13 · Word problem Five more than twice a number is 29. Find the number.
Let the number be x. Write 2x + 5 = 29. Then 2x = 24, so x = 12.
Q14 · Word problem The sum of two consecutive integers is 73. Find them.
Let them be x and x + 1. Then 2x + 1 = 73, so 2x = 72 and x = 36.
Q15 · Quadratic Solve: x² − 7x + 12 = 0.
Factorise: (x − 3)(x − 4) = 0.
Q16 · Intermediate Simplify: 5a − 2b + 3a + 7b.
Combine a-terms: 5a + 3a = 8a. Combine b-terms: −2b + 7b = 5b.
Q17 · Factorisation Factorise: 6x² + 9x.
Take out the common factor 3x.
Q18 · Equation Solve: 7 − 2x = 15.
Subtract 7: −2x = 8. Divide by −2.
Check: 7 − 2(−4) = 15.
Q19 · Identity Evaluate 101² using an identity.
Write 101 = 100 + 1.
Q20 · Mixed If 2x + 3 = 17, find 3x − 1.
First solve: 2x = 14, so x = 7. Substitute into the requested expression: 3(7) − 1 = 20.
14. Frequently Asked Questions
What is the easiest way to learn algebra?
Start with variables, coefficients, like terms, and the distributive property. Then move to linear equations, identities, factorisation, and word problems.
Which algebra topics should I prepare for competitive exams?
Study the topics listed in your target exam syllabus. Common foundations include simplification, algebraic identities, factorisation, linear equations, exponents, and translating word problems into equations.
How can I solve algebra questions faster?
Recognise patterns, combine like terms early, use identities where appropriate, factorise before expanding when useful, and check answers by substitution.
Do I need to memorise every algebraic identity?
Learn the most common identities and understand how they are derived. Understanding the pattern makes it easier to remember and apply them correctly.
Is algebra useful outside competitive examinations?
Yes. Algebra is a foundational part of mathematics used in science, technology, finance, data analysis, engineering, and everyday problem-solving.