a Plus b Plus c Whole Square Formula

a plus b plus c whole square formula

The formula of (a+b+c)² is utilized to determine the result the of sum of squares involving three numbers without eventually evaluating the squares.  

The formula of a plus b plus c whole square is a major algebraic identity. To get the expanded form of (a+b+c)² formula all you need to do is multiply (a+b+c) twice to determine the whole square of a + b + c. 

What is a plus b plus c whole square formula?

We just came to know that if we multiply the expression (a+b+c) twice we can get the (a+b+c)² formula quite easily. 

(a+b+c)² = (a+b+c) . (a+b+c)

or, (a+b+c)² = a² + ac + ab + ba + bc + b² + ca + c² + cb

Rearranging the right hand side of this equation we get:

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

or, (a+b+c)² = a² + b² + c² +2(ab +bc +ac)

Therefore the standard formula of a plus b plus c whole square is: 

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

Solved Examples

1. Find the exact value of a² + b² + c² when a + c + b = – 3.

Also, 1/b + 1/a + 1/c = 3 & abc = 4.

Solution: In this problem we have got three equations:

a + c + b = – 3 … (i)

1/b + 1/a + 1/c = 3 … (ii)

abc = 4 … (iii)

If we multiply equation (ii) with equation (iii)

abc . (1/b + 1/a + 1/c) = 4 x 3

We will have (ac + bc + ab) = 12 … (iv)

At this stage use the a plus b plus c whole square formula.

(a+b+c)² = a² + b² + c² +2(ab +bc +ca)

Putting the values from equation (i) and (iv) 

(-3)² = a² + b² + c² + 2 x 12

Therefore, a² + b² + c² = 9 – 24 = -15

2. Solve (4x + 5y + 3z)²

Solution: The expression (4x + 5y + 3z)² resembles the identity of (a+b+c)²

Here, a = 4x, b = 5y and c = 3z

Now put the values replacing a, b and c in the a plus b plus c whole square formula.

(4x + 5y + 3z)² = (4x)² + (5y)² + (3z)² + 2.4x.5y + 2.5y.3z + 2.3z.4x

or, (4x + 5y + 3z)² = 16x² + 25y² + 9z² + 40xy + 30yz + 24xz