01) What is Algebra & The Important Algebra Formulas For Competitive Exams?
Algebra is a branch of mathematics that uses symbols and letters (usually x, y, a, b) to represent numbers in formulas and equations. It helps us solve problems involving unknown quantities.
In competitive exams, algebra questions are usually formula-based, and with a little practice, they can be solved in seconds.
02) Important Algebra Formulas for Competitive Exams:
Here is a list of basic and advanced algebra formulas that frequently appear in exams.
Basic Identities:
| Formula No. | Formula |
|---|---|
| 1 | (a + b)² = a² + 2ab + b² |
| 2 | (a – b)² = a² – 2ab + b² |
| 3 | a² – b² = (a + b)(a – b) |
| 4 | (x + a)(x + b) = x² + (a + b)x + ab |
| 5 | (x – a)(x – b) = x² – (a + b)x + ab |
These five are the most used algebra formulas for competitive exams.
Cube Identities:
| Formula No. | Formula |
|---|---|
| 6 | (a + b)³ = a³ + b³ + 3ab(a + b) |
| 7 | (a – b)³ = a³ – b³ – 3ab(a – b) |
| 8 | a³ + b³ = (a + b)(a² – ab + b²) |
| 9 | a³ – b³ = (a – b)(a² + ab + b²) |
Special Products:
| Formula No. | Formula |
|---|---|
| 10 | (a + b + c)² = a² + b² + c² + 2(ab + bc + ca) |
| 11 | a² + b² = (a + b)² – 2ab |
| 12 | a² + b² + c² = (a + b + c)² – 2(ab + bc + ca) |
03) Examples Using Algebra Formulas for Competitive Exams:
Let’s look at how to apply these formulas with some quick examples.
Example 1:
Find the value of (a + b)² if a = 5 and b = 3.
Solution:
Using (a + b)² = a² + 2ab + b²
= 5² + 2×5×3 + 3²
= 25 + 30 + 9 = 64
Example 2:
Simplify: a³ + b³ if a = 2 and b = 3
Using the formula:
a³ + b³ = (a + b)(a² – ab + b²)
= (2 + 3)(4 – 6 + 9)
= 5 × 7 = 35
Example 3:
If a – b = 3 and ab = 40, find a² + b².
Use the identity: a² + b² = (a – b)² + 2ab
= 9 + 80 = 89
04) Practice Questions (with Answers):
Here are 10 algebra practice questions for competitive exams with detailed solutions:
Q1. If (x + y) = 7 and xy = 10, find x² + y².
Answer:
x² + y² = (x + y)² – 2xy
= 49 – 20 = 29
Q2. If a = 6, b = 4, find a² – b².
Answer:
a² – b² = (a + b)(a – b) = (10)(2) = 20
Q3. Find the value of (3x + 4)²
Answer:
= 9x² + 2×3x×4 + 16
= 9x² + 24x + 16
Q4. If a + b = 8 and ab = 15, find a³ + b³
Answer:
a³ + b³ = (a + b)³ – 3ab(a + b)
= 512 – 360 = 152
Q5. Simplify: (x + y)³ – (x – y)³
Answer:
Use identity:
(a + b)³ – (a – b)³ = 2b(3a² + b²)
= 2y(3x² + y²)
Q6. If x = 2 and y = -1, find (x + y)²
Answer:
= (2 – 1)² = 1
Q7. Find cube of (a – b) if a = 5 and b = 2
Answer:
(5 – 2)³ = 3³ = 27
Q8. Find a³ – b³ if a = 4 and b = 2
Answer:
= (a – b)(a² + ab + b²)
= (2)(16 + 8 + 4) = 2 × 28 = 56
Q9. If (x – y) = 5 and xy = 6, find x² + y²
Answer:
= (x – y)² + 2xy = 25 + 12 = 37
Q10. If a + b + c = 6, ab + bc + ca = 11, find a² + b² + c²
Answer:
= (a + b + c)² – 2(ab + bc + ca)
= 36 – 22 = 14
05) Algebra Formula Chart for Quick Revision:
| Type | Formula |
|---|---|
| Square of sum | (a + b)² = a² + 2ab + b² |
| Square of difference | (a – b)² = a² – 2ab + b² |
| Difference of squares | a² – b² = (a + b)(a – b) |
| Cube of sum | (a + b)³ = a³ + b³ + 3ab(a + b) |
| Cube of difference | (a – b)³ = a³ – b³ – 3ab(a – b) |
| Sum of cubes | a³ + b³ = (a + b)(a² – ab + b²) |
| Difference of cubes | a³ – b³ = (a – b)(a² + ab + b²) |
| Three-variable square | (a + b + c)² = a² + b² + c² + 2(ab + bc + ca) |
06) Tips to Remember Algebra Formulas for Competitive Exams:
- Understand, don’t memorize blindly – Know how each formula is derived.
- Practice daily – The more questions you solve, the more confident you’ll get.
- Create flashcards – Keep the algebra formulas for competitive exams handy.
- Take timed quizzes – To improve your speed and accuracy.
📥 Download PDF Version
To get a printable “Algebra Formulas for Competitive Exams PDF”, click here and revise anytime!
✨ Final Thoughts
Learning algebra becomes easy when you practice the right way. The algebra formulas for competitive exams mentioned above cover all the essential identities, shortcuts, and tricks needed to score high in quantitative aptitude sections.
Regular practice with real exam-style questions will help you stay ahead. Bookmark this page and revise before your next mock test!
