# 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.

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.

### 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.