Algebraic Identities
ICSE Class 8 · Mathematics · Chapter 12
What this chapter covers
- Identities and equations
- The square of a sum: (a + b)²
- The square of a difference: (a − b)²
- The product of a sum and a difference: (a + b)(a − b)
- The product (x + a)(x + b)
- Using identities for calculation
- Common errors with identities
Key points
- An algebraic identity is true for every value of its variables; an equation is true only for some values.
- (a + b)2 = a2 + 2ab + b2.
- (a - b)2 = a2 - 2ab + b2; the last term b2 is added, never subtracted.
- (a + b)(a - b) = a2 - b2, the difference of two squares.
- (x + a)(x + b) = x2 + (a + b)x + ab; add the numbers for the middle coefficient and multiply them for the last term.
- Each identity can be proved by multiplying out the brackets and collecting like terms.
- The letters in an identity can be replaced by any numbers or terms, such as 3x or 2y, used in the same way on both sides.
- Identities make mental arithmetic quick: write a number as a sum or difference of easy numbers, such as 98 = 100 - 2.
- a2 + b2 = (a + b)2 - 2ab = (a - b)2 + 2ab.
- (a + b)2 is never simply a2 + b2; the middle term 2ab must be included.
Questions answered
Is 4(x - 1) = 4x - 4 an identity or an equation? Give a reason.
It is an identity, because multiplying out the left side gives the right side, so it is true for every value of x.
Expand (m + 6)2.
m2 + 12m + 36.
Expand (4x - 3y)2.
16x2 - 24xy + 9y2.
Find the product (7a + 2b)(7a - 2b).
49a2 - 4b2.
Find the product (x - 8)(x + 3).
Here a + b = -5 and ab = -24, so the product is x2 - 5x - 24.
Use an identity to find 9992.
(1000 - 1)2 = 1000000 - 2000 + 1 = 998001.
Use an identity to find 62 × 58.
(60 + 2)(60 - 2) = 3600 - 4 = 3596.
Find the value of 752 - 252.
(75 + 25)(75 - 25) = 100 × 50 = 5000.