Formula for (a + b + c) Whole Square

One of the fundamental polynomial identities in basic algebra is the square of a trinomial, which is mathematically represented as $(a + b + c)^2$. Knowing how to expand this helps in geometry, quadratic equations, and calculus.

Question (Click to Flip)

What is the formula for (a - b - c)^2?

Answer

Following the same pattern but substituting negative signs: (a - b - c)^2 = a^2 + b^2 + c^2 - 2ab + 2bc - 2ca.

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Key Facts

It is known as the Square of a Trinomial.

The formula applies to real numbers, complex numbers, and matrices (if commutative).

If any variable is negative (like $a - b + c$), the sign of the product terms ($2ab$, $2bc$, etc.) changes accordingly.

1. The Expansion Formula

The standard algebraic formula is:

$(a + b + c)^2 = a^2 + b^2 + c^2 + 2ab + 2bc + 2ca$

This can also be written by factoring out the 2:

$(a + b + c)^2 = a^2 + b^2 + c^2 + 2(ab + bc + ca)$

2. How is it Derived?

You can derive this formula by multiplying the trinomial by itself:

$(a + b + c)(a + b + c)$
$= a(a+b+c) + b(a+b+c) + c(a+b+c)$
$= a^2 + ab + ac + ba + b^2 + bc + ca + cb + c^2$
Grouping the common terms together ($ab=ba$, $ac=ca$, $bc=cb$), we get the final formula.

Questions and Answers

What is the formula for (a - b - c)^2?+

Following the same pattern but substituting negative signs: (a - b - c)^2 = a^2 + b^2 + c^2 - 2ab + 2bc - 2ca.

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