# Formula and derivation of a minus b whole square explained

Learn and know the formula of a minus b whole square in algebra. a minus b whole square is also called as an identity. This formula is one of the most important formula in algebra.

Along with the formula mentioning I have given derivation of ${ \left( a-b \right) }^{ 2 }$ formula also. We can derive the formula in two different methods. First method is from the regular process that is give below. The second method is from ${ \left( a+b \right) }^{ 2 }$formula we can derive ${ \left( a-b \right) }^{ 2 }$ formula. For that what we need to do is in + sign you replace it with – sign.

## Derivation of a minus b whole square formula:

${ \left( a-b \right) }^{ 2 }$ = (a-b) × (a-b)

= a× (a-b) – b× (a-b)

=${ a }^{ 2 }$ – ab –ab + ${ b }^{ 2 }$

= ${ a }^{ 2 }$ -2ab + ${ b }^{ 2 }$

Therefore, ${ \left( a-b \right) }^{ 2 }$ = ${ a }^{ 2 }$ -2ab + ${ b }^{ 2 }$

a minus b whole square formula as follows:

a minus b whole square is equal to a square (${ a }^{ 2 }$) minus (-) product of 2, a and b plus (+) b square (${ b }^{ 2 }$)

i.e. ${ \left( a-b \right) }^{ 2 }$ = ${ a }^{ 2 }$ -2ab + ${ b }^{ 2 }$

${ \left( a-b \right) }^{ 2 }$+ 2ab = ${ a }^{ 2 }$ + ${ b }^{ 2 }$

Note:

a minus b whole square in terms of a plus b whole square formula.

${ \left( a-b \right) }^{ 2 }$ = ${ \left( a+b \right) }^{ 2 }$ – 4ab

Example:

Find ${ \left( a-b \right) }^{ 2 }$ value if a = 7 and b = 3 by using the formula.

Solution:

We know that,

${ \left( a-b \right) }^{ 2 }$ = ${ a }^{ 2 }$ -2ab + ${ b }^{ 2 }$

${ \left( a-b \right) }^{ 2 }$ = ${ 7 }^{ 2 }$ -2.7.3 + ${ 3 }^{ 2 }$

${ \left( a-b \right) }^{ 2 }$ = 49 – 42 + 9

${ \left( a-b \right) }^{ 2 }$ = 58 – 42

${ \left( a-b \right) }^{ 2 }$ = 16

Find ${ a }^{ 2 }$ + ${ b }^{ 2 }$ value if a-b = 3 and ab = 5.

Solution:

We know that

${ \left( a-b \right) }^{ 2 }$+ 2ab = ${ a }^{ 2 }$ + ${ b }^{ 2 }$

${ 3 }^{ 2 }$ + 2. 5 = ${ a }^{ 2 }$ + ${ b }^{ 2 }$

9 + 10 = ${ a }^{ 2 }$ + ${ b }^{ 2 }$

19 = ${ a }^{ 2 }$ + ${ b }^{ 2 }$

Find ${ \left( a-b \right) }^{ 2 }$ value if a+b = 6 and ab = 2.

Solution:

We know that,

${ \left( a-b \right) }^{ 2 }$ = ${ \left( a+b \right) }^{ 2 }$4ab

${ \left( a-b \right) }^{ 2 }$ = ${ 6 }^{ 2 }$ – 4.2

${ \left( a-b \right) }^{ 2 }$ = 36 – 8

${ \left( a-b \right) }^{ 2 }$ = 28