Students often get confused between different algebraic identities in Class 9 Maths. One of the most searched identities is the a2+b2 formula because it is used in simplification, polynomial questions, algebraic expressions, and board exam problems. Many students remember the formula for some time but forget how to apply it correctly in questions.
The good thing is that the a2+b2 formula is actually very easy once you understand the logic behind it. In this page, you will learn the formula, proof, solved examples, shortcut tricks, common mistakes, and important practice questions in a simple way.
If you are preparing for CBSE Class 9 Maths, this topic is very important from both school exams and revision point of view.
a2+b2 Formula
The formula is written as:
| Formula | Meaning |
|---|---|
| a² + b² = (a + b)² − 2ab | Sum of squares identity |
This identity helps students convert a square expression into another usable form for solving algebraic problems.
The a2+b2 formula is part of algebraic identities class 9 and is commonly used in simplification and polynomial chapters.
Understanding the a2+b2 Formula in Simple Language
In the a2+b2 formula, the terms a² and b² represent the squares of two numbers or variables.
For example:
- a² means a × a
- b² means b × b
When these two square terms are added together, the expression can also be written using another identity:
| Expression | Equivalent Form |
|---|---|
| a² + b² | (a + b)² − 2ab |
This identity becomes useful when students already know the values of:
- a + b
- ab
Instead of solving the entire expression directly, the formula helps in finding the answer faster.
Example
Suppose:
- a + b = 10
- ab = 21
Using the a2+b2 formula:
a² + b² = (a + b)² − 2ab
= 10² − 2 × 21
= 100 − 42
= 58
This is why the identity is important in algebraic simplification questions.
a2+b2 Formula Proof
Students usually memorize formulas but forget how they are formed. Understanding the proof helps in remembering the identity for a long time.
We know the identity:
(a + b)² = a² + 2ab + b²
Now rearrange the terms.
a² + b² = (a + b)² − 2ab
Hence proved.
This is the simplest derivation of the a2+b2 formula proof and it is enough for Class 9 exam preparation.
Why Students Confuse This Formula
Many students mix the a2+b2 formula with the formula of:
(a + b)² = a² + 2ab + b²
The confusion happens because both formulas contain similar terms.
Use this comparison table for quick revision.
| Formula | Expansion |
|---|---|
| (a + b)² | a² + 2ab + b² |
| (a − b)² | a² − 2ab + b² |
| a² + b² | (a + b)² − 2ab |
Remember this carefully:
- a² + b² does NOT contain the middle term directly
- The value of 2ab must be subtracted from (a + b)²
This is one of the most common mistakes in exams.
Solved Examples of a2+b2 Formula
These examples are designed according to CBSE Class 9 level.
Example 1
Find the value of a² + b² if:
- a + b = 7
- ab = 10
Solution
Using the a2+b2 formula:
a² + b² = (a + b)² − 2ab
= 7² − 2 × 10
= 49 − 20
= 29
Answer: 29
Example 2
If a + b = 12 and ab = 35, find a² + b².
Solution
a² + b² = (a + b)² − 2ab
= 12² − 2 × 35
= 144 − 70
= 74
Answer: 74
Example 3
Find the value of:
9² + 4²
Solution
= 81 + 16
= 97
Answer: 97
Example 4
If x + y = 15 and xy = 26, find x² + y².
Solution
Using the a2+b2 formula:
x² + y² = (x + y)² − 2xy
= 15² − 2 × 26
= 225 − 52
= 173
Answer: 173
Example 5
Find the value of a² + b² when:
- a + b = 20
- ab = 96
Solution
a² + b² = (a + b)² − 2ab
= 20² − 2 × 96
= 400 − 192
= 208
Answer: 208
Easy Trick to Remember the Formula
Most students forget where the minus sign comes in the identity.
Use this memory trick:
“Square first, subtract double product later”
Meaning:
- Find (a + b)²
- Subtract 2ab
That is the complete logic behind the a2+b2 formula.
You can also remember this pattern:
| Step | Action |
|---|---|
| Step 1 | Square the sum |
| Step 2 | Subtract 2ab |
| Final Answer | a² + b² |
This shortcut helps during quick revision before exams.
Common Mistakes Students Make
These mistakes are very common in school exams and MCQs.
Mistake 1
Writing:
a² + b² = (a + b)²
This is incorrect because the 2ab term is missing.
Mistake 2
Forgetting to subtract 2ab.
Correct formula:
a² + b² = (a + b)² − 2ab
Mistake 3
Calculation mistakes while squaring numbers.
Example:
12² = 144
Many students write incorrect square values during fast solving.
Mistake 4
Using the wrong sign.
The formula uses subtraction, not addition.
Practice Questions on a2+b2 Formula
Try these questions yourself before checking the answers.
- If a + b = 9 and ab = 14, find a² + b².
- Find x² + y² when x + y = 11 and xy = 24.
- If a + b = 18 and ab = 45, calculate a² + b².
- Find the value of 6² + 8².
- If p + q = 13 and pq = 30, find p² + q².
- Find a² + b² when:
- a + b = 25
- ab = 144
- Calculate:
- 15² + 7²
- If m + n = 16 and mn = 55, find m² + n².
MCQs on a2+b2 Formula
1. What is the correct identity for a² + b²?
A. (a + b)² + 2ab
B. (a + b)² − 2ab
C. (a − b)² − 2ab
D. a² − b²
Answer: B
2. Find a² + b² if a + b = 6 and ab = 8.
A. 16
B. 18
C. 20
D. 22
Answer: C
Solution
a² + b² = (a + b)² − 2ab
= 6² − 2 × 8
= 36 − 16
= 20
3. Which chapter contains this identity in Class 9?
A. Geometry
B. Statistics
C. Algebraic Identities
D. Coordinate Geometry
Answer: C
4. Find the value of 5² + 12².
A. 169
B. 144
C. 119
D. 194
Answer: A
5. If x + y = 14 and xy = 40, then x² + y² equals:
A. 96
B. 116
C. 124
D. 140
Answer: B
Quick Revision Notes
| Topic | Important Point |
|---|---|
| Formula | a² + b² = (a + b)² − 2ab |
| Chapter | Algebraic Identities |
| Most Common Mistake | Forgetting minus 2ab |
| Important For | CBSE Class 9 Maths |
| Used In | Simplification and polynomials |
Revision of a2+b2 Formula
Before exams, remember these three things clearly:
- a² + b² = (a + b)² − 2ab
- Do not confuse it with (a + b)²
- Always subtract the value of 2ab
Practice a few questions daily and this identity will become very easy to apply in exams.