a plus b whole square formula explained

Learn and know the formula of a plus b whole square in algebra. This is the first formula that we are going to learn in algebra chapter.

a plus b whole square formula explained

If we learn this formula, we can easily get the formula for a minus b whole square. Still one more formula also we can get i.e. a square plus b square ( { a }^{ 2 } + { b }^{ 2 } ). So now we will learn what is a plus b whole square formula along with that we will also learn the proof for this formula.

a plus b whole square formula as follows:

a plus b whole square mathematically written as { \left( a+b \right) }^{ 2 } . a plus b whole square { \left( a+b \right) }^{ 2 } is equal to the a square ( { a }^{ 2 } ) plus the product of 2, a and b (2ab) plus b square ( { b }^{ 2 } ).

Therefore, the formula of { \left( a+b \right) }^{ 2 } = { a }^{ 2 } + 2 × a × b + { b }^{ 2 } . This is one of the identities in algebra.

a plus b whole square formula explained

Derivation (proof) of a plus b whole square formula:

Along with knowing the formula of a plus b whole square, we should also know how we got that formula i.e. proof.

We know that we can write,

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

                  = × a + a × b + b × a + b × b (use multiplication of binomial with another binomial concept)

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

Therefore, { \left( a+b \right) }^{ 2 } = { a }^{ 2 } + 2 × a × b + { b }^{ 2 }

Form the above formula, we can get { a }^{ 2 } + { b }^{ 2 } = { \left( a+b \right) }^{ 2 } – 2 × a × b.

Note:

On this formula we can do one mathematics project also.

Examples:

Solve { \left( 3x+5y \right) }^{ 2 }

Solution:

We know that the given problem is looking like a plus b whole square formula.

So   { \left( a+b \right) }^{ 2 } = { a }^{ 2 } + 2 × a × b + { b }^{ 2 }

In this problem, we know that a = 3x and b = 5y

Therefore

{ \left( 3x+5y \right) }^{ 2 }    

= { (3x) }^{ 2 } + 2 × 3x × 5y + { (5y) }^{ 2 }

= 9 { x }^{ 2 } + 30xy + 25 { y }^{ 2 }

Therefore, { \left( 3x+5y \right) }^{ 2 } = 9 { x }^{ 2 } + 30xy + 25 { y }^{ 2 }

Solve: { \left( 4a+3 \right) }^{ 2 }

Solution:

According to the { \left( a+b \right) }^{ 2 } formula,

{ \left( 4a+3 \right) }^{ 2 }

= { (4a) }^{ 2 } + 2 × 4a × 3 + { 3 }^{ 2 }

= 16 { a }^{ 2 } + 24a + 9

Therefore, { \left( 4a+3 \right) }^{ 2 } = 16 { a }^{ 2 } + 24a + 9.

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