Square of Three Variables with Identity Expansion
Square of three variables means expanding an expression such as (a + b + c)². The expansion contains the squares of all three variables and twice each pairwise product. This page covers the standard identity, sign variations, calculation methods, shortcuts and solved examples used in algebra and quantitative aptitude.
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What Is the Square of Three Variables?
The standard formula is (a + b + c)² = a² + b² + c² + 2ab + 2bc + 2ca. There are three square terms and three pairwise product terms. For example, (x + 2 + y)² = x² + y² + 4 + 2xy + 4x + 4y.
Square of Three Variables Video Lessons
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Three-Variable Square Identity
Learn the three-variable square identity and understand how to apply the expression A² + B² + C² − AB − BC − CA in algebraic simplification.
Square of Three Variables Formula & Tricks
Important Formulas
Square each term and add twice the product of each of the three pairs.
The pairwise product signs follow the signs of the original terms.
This form handles any combination of plus and minus signs. The term 2q rbc is written with the product of q and r.
Multiplying each term in the first bracket by every term in the second gives the three square terms and six cross-products, which combine into three doubled products.
Quick Tricks
After writing the three square terms, list the pairs ab, bc and ca. Multiply each by 2 and apply the signs of the corresponding terms.
If two variables have a simple sum, first use (a + b + c)² = [(a + b) + c]². This can reduce arithmetic.
For an identity, put a = b = c = 1. The left side and expanded right side must both equal 9.
Square of Three Variables Concepts
Expansion of (a + b + c)²
The three pairs are (a, b), (b, c) and (c, a). Therefore, the cross-products are 2ab, 2bc and 2ca. No product of all three variables occurs because the expression has power 2.
Sign rule for mixed terms
For (a - b - c)², ab, bc and ca have signs negative, positive and negative respectively. Hence, (a - b - c)² = a² + b² + c² - 2ab + 2bc - 2ca.
Numerical evaluation using the identity
For positive values, direct addition may be shorter. For algebraic terms, the identity avoids repeated multiplication and keeps the pairwise products visible.
Finding a missing expression
From (a + b + c)² = a² + b² + c² + 2(ab + bc + ca), it follows that a² + b² + c² = (a + b + c)² - 2(ab + bc + ca).
Square of Three Variables Data Table
Use the structured data below while following the explanation and examples.
Common sign patterns
The signs of the cross-products depend on the signs of the original three terms.
| Expression | Expansion |
|---|---|
| (a + b + c)² | a² + b² + c² + 2ab + 2bc + 2ca |
| (a + b - c)² | a² + b² + c² + 2ab - 2bc - 2ca |
| (a - b + c)² | a² + b² + c² - 2ab - 2bc + 2ca |
| (a - b - c)² | a² + b² + c² - 2ab + 2bc - 2ca |
Note: The square terms always have positive signs; only the cross-product signs change.
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Square of Three Variables: Revision Points
Remember the identity, pairwise-product method and sign rules.
- (a + b + c)² = a² + b² + c² + 2ab + 2bc + 2ca.
- There are three square terms and three doubled pairwise-product terms.
- For signed terms, the sign of each cross-product is the product of the signs of its two variables.
- The square terms remain positive even when the original variables have negative signs.
- a² + b² + c² = (a + b + c)² - 2(ab + bc + ca).
- No abc term appears in the square of three variables.
Square of Three Variables FAQs
What is the square of three variables formula?
The formula is (a + b + c)² = a² + b² + c² + 2ab + 2bc + 2ca.
How do you expand (x + y - z)²?
(x + y - z)² = x² + y² + z² + 2xy - 2yz - 2zx.
What is (2 + 3 + 4)² using the three-variable identity?
(2 + 3 + 4)² = 4 + 9 + 16 + 12 + 24 + 16 = 81.
Why does no abc term occur in (a + b + c)²?
The expression has power 2, so multiplication produces products of two terms only. A product containing all three variables appears in a cube, not a square.
If a + b + c = 12 and ab + bc + ca = 35, what is a² + b² + c²?
Use a² + b² + c² = (a + b + c)² - 2(ab + bc + ca). Thus, the value is 12² - 2(35) = 74.