Square of Difference Identity with Algebraic Examples
Square of Difference Identity expresses the square of a binomial with a minus sign as three terms. The formula (a − b)² = a² − 2ab + b² helps expand algebraic expressions, simplify calculations and solve aptitude problems involving squares of differences.
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What Is the Square of Difference Identity?
The middle term is always negative when the expression is written as (a − b)². For example, (x − 5)² = x² − 10x + 25 because x² − 2(x)(5) + 5² gives x² − 10x + 25.
Square of Difference Identity Video Lessons
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Square Identities in Algebra
Understand the identities (A+B)², (A−B)², and A²−B², including their expansions and factorisation forms for solving algebraic expressions.
Square of Difference Identity Formula & Tricks
Important Formulas
Expand the square by squaring both terms and subtracting twice their product.
This is a different identity from the square of difference. It factors the difference between two squares.
An expression is a perfect square when it has two squared terms and a middle term equal to minus twice their product.
Quick Tricks
The terms in (a − b)² follow the pattern plus, minus, plus: a² − 2ab + b². Only the middle term is negative.
Write a number as a convenient larger number minus a difference, then apply the identity.
For A² − 2AB + B², verify that the middle term is minus twice the product of the square-root terms.
Square of Difference Identity Concepts
Expansion of Two Algebraic Terms
Apply the identity directly without multiplying the binomial twice. For (3x − 4)², take a = 3x and b = 4: (3x)² − 2(3x)(4) + 4² = 9x² − 24x + 16.
Numerical Square of a Difference
Choose a and b so that the calculation is simple. For 49², use 49 = 50 − 1. Therefore, 49² = 50² − 2(50)(1) + 1² = 2500 − 100 + 1 = 2401.
Recognising a Perfect-Square Trinomial
First identify the square roots of the first and last terms. Then check whether the middle term equals minus twice their product. Thus, 16x² − 40x + 25 = (4x − 5)² because −2(4x)(5) = −40x.
Difference Between Similar Identities
The square of difference is (a − b)² = a² − 2ab + b², while the difference of squares is a² − b² = (a − b)(a + b). The first produces three terms after expansion; the second factors into two binomials.
Square of Difference Identity Data Table
Use the structured data below while following the explanation and examples.
Common Forms of the Identity
Equivalent and related algebraic forms used in expansion and factorisation.
| Form | Result | Use |
|---|---|---|
| (a − b)² | a² − 2ab + b² | Expand a squared difference |
| a² − 2ab + b² | (a − b)² | Factor a perfect-square trinomial |
| a² − b² | (a − b)(a + b) | Factor a difference of squares |
Note: Do not confuse (a − b)² with a² − b²; the former contains the additional terms −2ab and b².
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Square of Difference Identity Revision Points
Core formulas, sign rules and recognition checks for quick revision.
- (a − b)² = a² − 2ab + b².
- The middle term is negative and equals −2ab.
- The first and last terms are the squares of the two terms.
- For a perfect-square trinomial A² − 2AB + B², the factorised form is (A − B)².
- The identity a² − b² = (a − b)(a + b) is different from the square of difference identity.
- For quick numerical calculation, write a number as a convenient base minus its difference.
Square of Difference Identity FAQs
What is the square of difference identity formula?
The formula is (a − b)² = a² − 2ab + b².
What is the expansion of (x − 9)²?
(x − 9)² = x² − 2(x)(9) + 9² = x² − 18x + 81.
How do you calculate 96² using the identity?
Write 96 as 100 − 4. Then 96² = (100 − 4)² = 10000 − 800 + 16 = 9216.
How can x² − 14x + 49 be factorised?
Since 14x = 2(x)(7) and 49 = 7², the expression is (x − 7)².
Is (a − b)² equal to a² − b²?
No. (a − b)² = a² − 2ab + b², whereas a² − b² = (a − b)(a + b).
What is (5m − 2n)²?
(5m − 2n)² = 25m² − 20mn + 4n².