a2-b2 Formula

About a²-b² Formula

The a²- b² formula is a special type of algebraic identity. This formula is used to factorise the square of the difference of a binomial.

It is also known as "the difference of squares formula".The a²- b² is read as a square minus b square. More Maths Formulas on the parent's page.

Proof of a² - b² Formula

1. The a²- b² formula is given by
2. a²- b² =(a+b)(a-b), so let’s verify it
3. (a + b)(a - b) = (a² - b²) .???????
4. Proof
5. We have:
6. (a + b)(a - b) = a(a - b) + b(a - b)
7. (a + b)(a - b) =a² -ab+ba-b²
8. (a + b)(a - b) =a² -ab+ab-b² [since ba = ab]
9. (a + b)(a - b) =a² -b² .
10. Therefore,(a+b)(a-b)=(a² -b² ).

Solved Examples on a²-b² Formula

Let us solve some problems using the a²-b² formula.

1. Example 1 Find each of the following products:
   1. (4x + 5y)(4x - 5y)
   2. Solution
   3. We have:
   4. Using (a+b)(a-b)=(a² -b² )
   5. (4x + 5y)(4x - 5y)
   6. = (4x) ² - (5y) ²
   7. =(16x²-25y² )
2. Example 2:Factorise(x³ - x)
   1. SOLUTION
   2. We have
   3. (i) (x³ − x) = x(x² − 1)
   4. = x(x-1)(x + 1)
   5. [Using (a²-b²) = (a−b)(a + b)]
   6. (x³-x) = x(x-1)(x+1).
3. Example 3:Simplify 15²- 5²usinga²-b²formula
   1. Solution:To find15²- 5²
   2. Let us assume a = 15 and b = 5
   3. Using formulaa²- b²= (a - b) (a+ b)
   4. 15²-5²= (15 - 5) (15 + 5)
   5. = 15(15 +5) - 5(15 + 5)
   6. = 225+75 -75 -25
   7. Answer:15²-5²= 200
4. Example 4: Factorise x ^ 2 - 1 - 2a - a ^ 2
   1. SOLUTION
   2. We have
   3. x ² - 1 - 2a - a ²
   4. =x²-(1+2a+a² )
   5. =x² -(1+a)²
   6. [Using (a²-b²) = (a−b)(a + b)]
   7. =(x-1-a)(x+1+ a) .
   8. Therefore, (x ² - 1 - 2a - a ²) = (x - 1 - a)(x + 1 + a) 

Identity: An identity is equality which is true for all values of the variable(s).

Standard identities

Identity 1 :    (a + b)2 = a2 + 2ab + b2

Identity 2 :    (a – b)2 = a2 – 2ab + b2

Identity 3 :    (a + b) (a – b) = a2 – b2

Identity 4 :    (x + a) (x + b) = x2 + (a + b) x + ab

Identity 5 :    (a + b + c)2 = (a2 + b2 + c2) + 2(ab + bc + ca)

Identity 6 :    (a + b – c)2 = (a2 + b2 + c2) + 2(ab – bc - ca)

Identity 7 :    (a + b)3 = a3 + b3 + 3ab (a + b)

Identity 8 :    (a - b)3 = a3 – b3 – 3ab (a – b)

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