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.

Formula and derivation of a minus b whole square

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

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