(a + b)² Identity
The identity (a + b)² = a² + 2ab + b² is one of the most fundamental algebraic identities in mathematics. It gives the expansion of the square of the sum of two terms.
This identity is used extensively in simplification, factorisation, and mental mathematics. It is different from an equation because an identity is true for all values of the variables, not just specific values.
In Class 8 NCERT, this identity is introduced along with other standard identities. Mastering it is essential because it appears in higher classes in quadratic equations, polynomials, and coordinate geometry.
What is (a + b)² Identity?
Definition: An algebraic identity is an equation that is true for all values of the variables involved.
The (a + b)² identity states:
(a + b)² = a² + 2ab + b²
Key Terms:
- a and b are any real numbers or algebraic expressions.
- a² is the square of the first term.
- b² is the square of the second term.
- 2ab is twice the product of the two terms (the "middle term").
- The left side (a + b)² means (a + b) multiplied by itself: (a + b)(a + b).
Identity vs Equation:
- An identity is true for ALL values of variables. Example: (a + b)² = a² + 2ab + b² is true for a = 1, b = 2 or a = 5, b = 3 or any values.
- An equation is true only for specific values. Example: x + 3 = 7 is true only when x = 4.
(a + b)² Identity Formula
The (a + b)² Identity:
(a + b)² = a² + 2ab + b²
Where:
- a = first term
- b = second term
Useful rearrangements:
- a² + b² = (a + b)² - 2ab
- 2ab = (a + b)² - a² - b²
- (a + b) = √(a² + 2ab + b²) when a + b > 0
Common mistake:
- WRONG: (a + b)² = a² + b² (this misses the 2ab term)
- RIGHT: (a + b)² = a² + 2ab + b²
Derivation and Proof
Algebraic Proof:
- (a + b)² = (a + b)(a + b)
- Using distributive property (FOIL method):
- = a(a + b) + b(a + b)
- = a² + ab + ba + b²
- = a² + ab + ab + b²
- = a² + 2ab + b²
Consider a square with side length (a + b).
- The area of the square = (a + b)².
- Divide the square into 4 parts by drawing lines at distance 'a' from two adjacent sides.
- This creates:
- One square of side a with area = a² (top-left)
- One square of side b with area = b² (bottom-right)
- Two rectangles, each with dimensions a and b, area = ab each (top-right and bottom-left)
- Total area = a² + ab + ab + b² = a² + 2ab + b²
Numerical Verification:
Let a = 3, b = 2:
- LHS = (3 + 2)² = 5² = 25
- RHS = 3² + 2(3)(2) + 2² = 9 + 12 + 4 = 25
- LHS = RHS. Verified.
Types and Properties
The (a + b)² identity is applied in several types of problems:
1. Direct Expansion:
- Expand (x + 5)² = x² + 10x + 25
- Here a = x, b = 5
2. Evaluating Squares of Numbers (Mental Maths):
- Find 102² = (100 + 2)² = 10000 + 400 + 4 = 10404
3. Simplification:
- Simplify (2x + 3y)² = 4x² + 12xy + 9y²
4. Finding a² + b² when (a + b) and ab are known:
- Use a² + b² = (a + b)² - 2ab
5. Factorisation:
- Factorise x² + 6x + 9 = (x + 3)²
- Recognise the pattern: first term is a perfect square, last term is a perfect square, middle term = 2ab
6. With Algebraic Expressions:
- Expand (3a + 4b)² = 9a² + 24ab + 16b²
Solved Examples
Example 1: Example 1: Direct expansion
Problem: Expand (x + 7)².
Solution:
Using (a + b)² = a² + 2ab + b²:
- Here a = x, b = 7
- (x + 7)² = x² + 2(x)(7) + 7²
- = x² + 14x + 49
Answer: (x + 7)² = x² + 14x + 49
Example 2: Example 2: Evaluating a number squared
Problem: Find the value of 105² using the identity.
Solution:
- 105 = 100 + 5, so a = 100, b = 5
- 105² = (100 + 5)²
- = 100² + 2(100)(5) + 5²
- = 10000 + 1000 + 25
- = 11025
Answer: 105² = 11025
Example 3: Example 3: Expansion with algebraic terms
Problem: Expand (3p + 4q)².
Solution:
- Here a = 3p, b = 4q
- (3p + 4q)² = (3p)² + 2(3p)(4q) + (4q)²
- = 9p² + 24pq + 16q²
Answer: (3p + 4q)² = 9p² + 24pq + 16q²
Example 4: Example 4: Finding a² + b²
Problem: If a + b = 9 and ab = 14, find a² + b².
Solution:
Using: a² + b² = (a + b)² - 2ab
- a² + b² = 9² - 2(14)
- = 81 - 28
- = 53
Answer: a² + b² = 53
Example 5: Example 5: Factorisation using the identity
Problem: Factorise x² + 10x + 25.
Solution:
- x² = (x)² — so first term = x
- 25 = (5)² — so second term = 5
- Middle term check: 2(x)(5) = 10x ✔
- This matches the pattern a² + 2ab + b² = (a + b)²
Answer: x² + 10x + 25 = (x + 5)²
Example 6: Example 6: Mental calculation of 53²
Problem: Find 53² without direct multiplication.
Solution:
- 53 = 50 + 3
- 53² = (50 + 3)²
- = 50² + 2(50)(3) + 3²
- = 2500 + 300 + 9
- = 2809
Answer: 53² = 2809
Example 7: Example 7: Expansion with fractions
Problem: Expand (x + 1/2)².
Solution:
- a = x, b = 1/2
- (x + 1/2)² = x² + 2(x)(1/2) + (1/2)²
- = x² + x + 1/4
Answer: (x + 1/2)² = x² + x + 1/4
Example 8: Example 8: Finding (a + b) from a² + b² and ab
Problem: If a² + b² = 40 and ab = 12, find (a + b)² and (a + b).
Solution:
- (a + b)² = a² + 2ab + b²
- = (a² + b²) + 2ab
- = 40 + 2(12)
- = 40 + 24
- = 64
Therefore, a + b = √64 = 8 (taking positive value).
Answer: (a + b)² = 64, so a + b = 8.
Example 9: Example 9: Simplifying an expression
Problem: Simplify (2x + 3)² + (2x - 3)².
Solution:
Expanding each:
- (2x + 3)² = 4x² + 12x + 9
- (2x - 3)² = 4x² - 12x + 9
Adding:
- = 4x² + 12x + 9 + 4x² - 12x + 9
- = 8x² + 18
- = 2(4x² + 9)
Answer: (2x + 3)² + (2x - 3)² = 8x² + 18
Example 10: Example 10: Product using identity
Problem: Find the value of 103 x 103 using the identity.
Solution:
- 103 x 103 = 103² = (100 + 3)²
- = 100² + 2(100)(3) + 3²
- = 10000 + 600 + 9
- = 10609
Answer: 103 x 103 = 10609
Real-World Applications
Mental Mathematics: The identity allows quick calculation of squares of numbers close to round numbers. For example, 52² = (50 + 2)² = 2500 + 200 + 4 = 2704.
Factorisation: Recognising expressions of the form a² + 2ab + b² as perfect square trinomials helps in factorising algebraic expressions quickly.
Completing the Square: This identity is the basis for the "completing the square" method used in Class 10 to solve quadratic equations.
Geometry: The geometric proof connects algebra with area calculation. A square of side (a + b) has area that splits into four parts matching the identity.
Higher Mathematics: This identity extends to (a + b + c)² and is used in binomial theorem, polynomial expansion, and coordinate geometry.
Physics: Expressions of the form (v + u)² appear in kinematics and energy calculations, where this identity is used to expand and simplify.
Key Points to Remember
- (a + b)² = a² + 2ab + b² — this is the most important identity to memorise.
- It is true for ALL values of a and b (it is an identity, not an equation).
- Common mistake: (a + b)² ≠ a² + b². The 2ab term must NOT be forgotten.
- The middle term 2ab is always positive in (a + b)².
- To factorise a² + 2ab + b², write it as (a + b)².
- To find a² + b², use the rearrangement: a² + b² = (a + b)² - 2ab.
- The geometric proof uses area of a square of side (a + b) split into 4 parts.
- This identity is used for mental calculation of squares: n² = (nearest round number + difference)².
- Both a and b can be numbers, variables, or algebraic expressions.
- This identity forms the foundation for completing the square and the quadratic formula.
Practice Problems
- Expand (y + 9)².
- Find the value of 203² using the identity.
- Expand (5a + 2b)².
- If x + y = 12 and xy = 32, find x² + y².
- Factorise: 4x² + 12x + 9.
- Evaluate 998² using a suitable identity. (Hint: 998 = 1000 - 2, use (a - b)² or rewrite.)
- Simplify: (a + 3)² - (a - 3)².
- If a² + b² = 100 and ab = 48, find (a + b).
Frequently Asked Questions
Q1. What is (a + b)²?
(a + b)² = a² + 2ab + b². It is the expansion of the square of the sum of two terms a and b.
Q2. Why is (a + b)² not equal to a² + b²?
Because (a + b)² = (a + b)(a + b). When you multiply, you get the cross terms ab + ab = 2ab in addition to a² and b². So (a + b)² = a² + 2ab + b².
Q3. What is the difference between an identity and an equation?
An identity is true for all values of the variables (e.g., (a + b)² = a² + 2ab + b² for any a, b). An equation is true only for specific values (e.g., x + 3 = 7 only when x = 4).
Q4. How do you use this identity for mental maths?
Write the number as (round number + small number). For example, 52² = (50 + 2)² = 2500 + 200 + 4 = 2704. This avoids long multiplication.
Q5. How do you prove (a + b)² = a² + 2ab + b² geometrically?
Draw a square of side (a + b). Divide it into: one square of area a², one square of area b², and two rectangles of area ab each. Total = a² + 2ab + b².
Q6. What is the value of (a + b)² - (a - b)²?
(a + b)² - (a - b)² = (a² + 2ab + b²) - (a² - 2ab + b²) = 4ab.
Q7. How do you factorise using this identity?
If an expression matches the form a² + 2ab + b² (perfect square trinomial), it can be written as (a + b)². For example, x² + 8x + 16 = (x + 4)².
Q8. Can a and b be negative in (a + b)²?
Yes. The identity works for all real values. If b is negative, say b = -3, then (a + (-3))² = (a - 3)², which uses the (a - b)² identity instead.
Q9. How do you find a² + b² if you know a + b and ab?
Use a² + b² = (a + b)² - 2ab. For example, if a + b = 7 and ab = 10, then a² + b² = 49 - 20 = 29.
Q10. What is (a + b + c)²?
(a + b + c)² = a² + b² + c² + 2ab + 2bc + 2ca. This is the extended version of the (a + b)² identity for three terms.
