What are Algebraic Identities?
Algebraic identities are equations that are true for all values of the variables. They help us simplify complex algebraic expressions and perform calculations mentally.
The Three Standard Identities:
| Identity | Formula | Name |
|---|---|---|
| Identity I | \((a + b)^2 = a^2 + 2ab + b^2\) | Square of a Sum |
| Identity II | \((a - b)^2 = a^2 - 2ab + b^2\) | Square of a Difference |
| Identity III | \((a + b)(a - b) = a^2 - b^2\) | Difference of Squares |
Verification of Identities:
For Identity I: \((a + b)^2 = (a + b)(a + b) = a^2 + ab + ba + b^2 = a^2 + 2ab + b^2\)
Applications of Identities:
- Expanding expressions: \((3x + 5)^2 = (3x)^2 + 2(3x)(5) + 5^2 = 9x^2 + 30x + 25\)
- Factorizing expressions: \(x^2 - 9 = (x)^2 - (3)^2 = (x + 3)(x - 3)\)
- Simplifying calculations mentally: \(102^2 = (100 + 2)^2 = 100^2 + 2(100)(2) + 2^2 = 10000 + 400 + 4 = 10404\)
Mental Mathematics Using Identities:
These identities make it easy to calculate squares and products of large numbers mentally:
| Calculation | Using Identity | Mental Math |
|---|---|---|
| \(35^2\) | \((30 + 5)^2 = 900 + 300 + 25 = 1225\) | Add 300 to 900 to get 1200, plus 25 |
| \(28 \times 32\) | \((30 - 2)(30 + 2) = 900 - 4 = 896\) | Square of middle number minus square of difference |
| \(99^2\) | \((100 - 1)^2 = 10000 - 200 + 1 = 9801\) | Subtract 200 from 10000, add 1 |
More Useful Identities:
\((x + a)(x + b) = x^2 + (a + b)x + ab\)
\((a + b + c)^2 = a^2 + b^2 + c^2 + 2ab + 2bc + 2ca\)