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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