Difference of Squares Identity with Factorisation Steps
Difference of Squares Identity expresses the difference between two perfect squares as a product of their sum and difference. The standard formula is a² − b² = (a − b)(a + b). It helps factorise algebraic expressions, simplify calculations and solve aptitude problems involving squares.
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What Is the Difference of Squares Identity?
Here, a² and b² must be perfect square expressions, while the operation between them must be minus. For example, x² − 49 = x² − 7² = (x − 7)(x + 7). The identity does not apply directly to a² + b².
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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.
Difference of Squares Identity Formula & Tricks
Important Formulas
The difference of two squares factors into the product of their difference and their sum.
Using the distributive law, the middle terms −ab and +ab cancel.
This form is useful for quickly calculating the difference between two nearby squares.
Quick Tricks
The identity requires subtraction between the square terms. First rewrite each term as a square, then use difference of the bases and sum of the bases.
For two numbers that are close together, multiply their difference by their sum instead of calculating both squares separately.
An expression of the form (a − b)(a + b) can be reduced directly to a² − b².
Difference of Squares Identity Concepts
Identifying Perfect Square Terms
The expressions 25x², 49y⁴ and 100 are perfect squares because they can be written as (5x)², (7y²)² and 10² respectively. After this rewrite, factorise the expression using the difference of the bases and the sum of the bases.
Factorising a Difference of Squares
For a² − b², the factors are (a − b) and (a + b). The first terms in both factors are the square roots of the original square terms. The constant or variable part changes sign between the two factors.
Using the Identity for Fast Arithmetic
For N² − M², calculate N − M and N + M, then multiply the results. This avoids separately finding N² and M².
Solving Equations by Factorisation
If x² − c² = 0, then (x − c)(x + c) = 0. Therefore, x = c or x = −c. The same method applies when the square terms contain algebraic expressions.
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Difference of Squares Identity Revision Points
Recall the pattern, conditions and common applications of the identity.
- Use a² − b² = (a − b)(a + b).
- Both terms must be expressible as perfect squares.
- The operation must be subtraction, not addition.
- The two factors have the same first terms and opposite second terms.
- For fast arithmetic, use N² − M² = (N − M)(N + M).
- If x² − c² = 0, the solutions are x = c and x = −c.
- Do not factorise a² + b² using the real-number difference of squares identity.
Difference of Squares Identity FAQs
What is the difference of squares identity formula?
The formula is a² − b² = (a − b)(a + b).
How do you factorise x² − 64?
Since 64 = 8², x² − 64 = x² − 8² = (x − 8)(x + 8).
Can the identity be used for a² + b²?
No. The identity applies to a difference, a² − b². The sum a² + b² does not factorise into (a − b)(a + b).
What is 75² − 65² using the identity?
75² − 65² = (75 − 65)(75 + 65) = 10 × 140 = 1400.
How do you factorise 9x² − 25?
Write it as (3x)² − 5². Therefore, 9x² − 25 = (3x − 5)(3x + 5).
What are the solutions of x² − 36 = 0?
Factorising gives (x − 6)(x + 6) = 0. Hence, x = 6 or x = −6.