Square of Sum Identity with Expansion Examples
Square of Sum Identity gives the expansion of the square of two added terms. The formula is (a + b)² = a² + 2ab + b². It is used to simplify algebraic expressions, calculate squares quickly and solve quantitative aptitude problems involving perfect-square expressions.
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What Is the Square of Sum Identity?
Here, a and b may be numbers, variables or algebraic expressions. The middle term 2ab is positive because the two terms are added. For example, (x + 5)² = x² + 2(x)(5) + 5² = x² + 10x + 25.
Square of Sum 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 Sum Identity Formula & Tricks
Important Formulas
Square each term separately and add twice the product of the two terms.
The middle term is twice the product of the terms inside the bracket.
This is the same identity written with numerical terms m and n.
Quick Tricks
For (a + b)², write the square of the first term, twice the product, and the square of the second term.
Express a number as a convenient sum, then apply the identity instead of multiplying the full number directly.
The middle term must be 2ab, not ab. This check helps avoid the most common expansion error.
Square of Sum Identity Concepts
Expanding Algebraic Expressions
First square the first term, then multiply the terms and double the product, and finally square the second term. If either term contains a coefficient, include that coefficient in the multiplication.
Squaring a Binomial with Constants
The identity reduces long multiplication. Both terms are squared, while the product of the terms is doubled.
Using the Identity for Mental Calculation
Choose a convenient decomposition such as 100 + 4 or 50 + 3. Then calculate the two squares and the double product separately.
Recognising a Perfect Square
The first and last terms must be perfect squares, and the middle term must equal twice the product of their square roots. For positive terms, the resulting binomial has a plus sign.
Common Sign and Bracket Rules
Do not square only the first and last terms. In (a + b)², the cross-product terms ab and ba combine to give 2ab. Also, (−a + b)² can be treated as (b − a)², giving b² − 2ab + a².
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Square of Sum Identity Revision Points
Key formulas, checks and applications for quick revision.
- (a + b)² = a² + 2ab + b².
- The middle term is twice the product of the two terms.
- For (a + b)², all three expanded terms have positive signs.
- To identify a perfect square, check whether the middle term equals twice the product of the square roots of the first and last terms.
- Use a convenient sum such as 100 + 3 to calculate numerical squares quickly.
- Do not omit the cross-product term 2ab.
Square of Sum Identity FAQs
What is the square of sum identity formula?
The formula is (a + b)² = a² + 2ab + b².
How do you expand (2x + 5)²?
(2x + 5)² = (2x)² + 2(2x)(5) + 5² = 4x² + 20x + 25.
What is the value of 102² using the square of sum identity?
102² = (100 + 2)² = 10000 + 400 + 4 = 10404.
Why is the middle term in (a + b)² equal to 2ab?
Expanding gives (a + b)(a + b) = a² + ab + ab + b². The two ab terms combine to give 2ab.
How can x² + 16x + 64 be factorised using this identity?
Since 64 = 8² and 16x = 2(x)(8), x² + 16x + 64 = (x + 8)².
What is the difference between (a + b)² and a² + b²?
(a + b)² includes the cross-product term 2ab: (a + b)² = a² + 2ab + b². Therefore, it is generally not equal to a² + b².