Algebraic Identities
Algebraic Identities
Identities are the highest-return topic in the algebra chapter because they solve questions in one line that would otherwise take five. Expect 2 questions directly, and several more where an identity is the fastest route. There are only a handful to learn.
The Standard Identities
| Identity | Expansion |
|---|---|
| (a + b)² | a² + 2ab + b² |
| (a − b)² | a² − 2ab + b² |
| a² − b² | (a + b)(a − b) |
| (x + a)(x + b) | x² + (a + b)x + ab |
| (a + b)³ | a³ + b³ + 3ab(a + b) |
| (a − b)³ | a³ − b³ − 3ab(a − b) |
| a³ + b³ | (a + b)(a² − ab + b²) |
| a³ − b³ | (a − b)(a² + ab + b²) |
An identity is true for every value of the variables, unlike an equation, which is true only for particular values. That definitional difference is itself examined.
The Three-Term Identity
(a + b + c)² = a² + b² + c² + 2ab + 2bc + 2ca
This one is asked in reverse more often than forwards. If a + b + c and a² + b² + c² are given, the value of ab + bc + ca follows immediately from rearranging it.
If a + b + c = 0, then a³ + b³ + c³ = 3abc
That conditional result is a standard one-mark question and cannot be derived quickly in the hall, so it must be memorised.
Using Identities as Shortcuts
- Squaring near a round number: 98² = (100 − 2)² = 10000 − 400 + 4 = 9604.
- Difference of squares: 87² − 13² = (87 + 13)(87 − 13) = 100 × 74 = 7400.
- Product of near numbers: 103 × 97 = (100 + 3)(100 − 3) = 10000 − 9 = 9991.
Each of these takes about ten seconds by identity and close to a minute by long multiplication. Over forty questions that difference decides whether you attempt the paper fully.
The Reciprocal Family
If x + 1/x = k, then x² + 1/x² = k² − 2
If x − 1/x = k, then x² + 1/x² = k² + 2
Watch the sign: the same target expression takes minus 2 in one case and plus 2 in the other. Both wrong versions appear among the options, so read whether the given expression has a plus or a minus.
The cube extension follows the same logic: x³ + 1/x³ = k³ − 3k when x + 1/x = k.
✗ (a − b)² = a² − b² | ✓ (a − b)² = a² − 2ab + b²; a² − b² is a different identity
सर्वसमिका हर मान के लिए सत्य होती है, समीकरण केवल कुछ मानों के लिए।
TRE pointer: The most repeated confusion in the whole algebra chapter is between (a − b)² and a² − b², and it is exploited almost every year. Papers also favour the reciprocal family because it looks harder than it is. Since a teaching post is at stake, expect at least one question testing whether you know that an identity holds for all values while an equation holds only for some. Identities also raise your attempt rate, which matters directly when a wrong answer costs −1/3.
60-Second Recap
- An identity is true for all values; an equation only for particular values.
- (a+b)², (a−b)² and a²−b² are the three that appear most often.
- (a+b+c)² expands to the three squares plus twice each pair product.
- If a+b+c = 0 then a³+b³+c³ = 3abc.
- x + 1/x = k gives x² + 1/x² = k² − 2; the minus version gives k² + 2.
- Use identities for near-round arithmetic: 98², 103 × 97, 87² − 13².
Frequently Asked Questions
What is the difference between an identity and an equation?
An identity is true for every value of the variable, such as a plus b, all squared, equalling a squared plus two ab plus b squared. An equation is true only for particular values, such as two x plus three equals seven, which holds only when x is two.
If a plus b plus c is zero, what is a cubed plus b cubed plus c cubed?
It equals three abc. This is a conditional identity that cannot be derived quickly under exam pressure, so it must be memorised. Questions using it usually give you three numbers that sum to zero and ask for the sum of their cubes.
If x plus one over x equals five, what is x squared plus one over x squared?
Twenty-three. Square the given expression to get x squared plus two plus one over x squared, then subtract two. The general rule is k squared minus two. If the given expression had a minus sign instead, the answer would be k squared plus two.
Which identities are most useful for fast calculation?
The difference of squares and the two square identities. They turn 98 squared, 103 times 97 and 87 squared minus 13 squared into ten-second problems. Over a forty-question paper this is the difference between attempting everything and running out of time.