Whole Square Formula for a−b−ca-b-ca−b−c

The expression a−b−ca-b-ca−b−c can be treated as (a−(b+c))(a-(b+c))(a−(b+c)) when squaring it. The whole square formula (also known as the square of a sum or difference) is:

(x−y)2=x2−2xy+y2(x-y)^2=x^2-2xy+y^2(x−y)2=x2−2xy+y2

Applying this to a−b−ca-b-ca−b−c:

Rewrite as:

(a−(b+c))2(a-(b+c))^2(a−(b+c))2

Using the formula:

=a2−2a(b+c)+(b+c)2=a^2-2a(b+c)+(b+c)^2=a2−2a(b+c)+(b+c)2

Now expand:

=a2−2ab−2ac+(b2+2bc+c2)=a^2-2ab-2ac+(b^2+2bc+c^2)=a2−2ab−2ac+(b2+2bc+c2)

Final expanded form:

a2−2ab−2ac+b2+2bc+c2a^2-2ab-2ac+b^2+2bc+c^2a2−2ab−2ac+b2+2bc+c2

Summary:

(a−b−c)2=a2−2ab−2ac+b2+2bc+c2\boxed{(a-b-c)^2=a^2-2ab-2ac+b^2+2bc+c^2}(a−b−c)2=a2−2ab−2ac+b2+2bc+c2​

This is the whole square formula for the expression a−b−ca-b-ca−b−c.