# EXPANDING BRACKETS WITH POWERS OF 2

On this webpage "expanding brackets with powers of 2", we are going to see how to expand algebraic expressions of power 2.

## What is Algebraic identity?

An identity is an equality that remains true regardless of the values of any variables that appear within it.

## Expanding brackets with powers of 2 - Examples

Question 1 :

Expand (5x + 3)²

Solution :

Here the given question is in the form of (a+b)². Instead of a we have "5x" and instead of b we have "3" .

So we need to apply the formula a² + 2ab + b² and we need to apply those values of a and b

a = 5 x and b = 3

(5x + 3)² = (5x)² + 2 (5x) (3) + (3)²

= 25x² + 30 x + 9

= 25x² + 30 x + 9

Let us see the next example problem on "Expanding brackets with powers of 2". Question 2 :

Expand (x + 2)²

Solution :

Here the question is in the form of (a+b)². Instead of a we have "x" and instead of b we have "2".

So we need to apply the formula a² + 2ab + b ² and we need to apply those values of a and b

a = x   and b = 2

(x + 2)² = (x)² + 2 (x) (2) + (2)²

= x² + 4 x + 4

Let us see the next example problem on "Expanding brackets with powers of 2".

Question 3 :

(105)²  is equal to

Solution :

Instead of multiplying 105 x 105 to get the value of (105)² we can use algebraic formula for a plus b whole square that is  (a+b)² to get the same answer.105 can be written as 100 + 5.

(105)² = (100 + 5)²

(a + b)² = a² + b² + 2 a b

a = 100  b = 5

(105)² = (100)² + (5)² + 2 (100)(5)

= 10000 + 25 + 1000

= 11025

Let us see the next example problem on "Expanding brackets with powers of 2".

Question 4 :

Expand (2x - 3) ²

Solution :

Here the question is in the form of (a-b) ². Instead of a we have "2x" and instead of b we have "3" . Now we need to apply the formula a² - 2ab + b ² and we need to apply those values instead of a and b

(2x - 3)² = (2x)² - 2 (2x) (3) + (3)²

= 4x² - 12 x + 9

Let us see the next example problem on "Expanding brackets with powers of 2".

Question 5 :

Expand (x - 5) ²

Solution :

Here the question is in the form of (a-b) ². Instead of a we have "x" and instead of b we have "5" . Now we need to apply the formula a² - 2ab + b ² and we need to apply those values instead of a and b

(x - 5)² = (x)² - 2 (x) (5) + (5)²

= x² - 10 x + 25

Let us see the next example problem on "Expanding brackets with powers of 2". Question 6 :

Expand (3x - 7) ²

Solution :

Here the question is in the form of (a-b) ². Instead of a we have "3x" and instead of b we have "7" . Now we need to apply the formula for a minus b whole square that is  a² - 2ab + b ² and we need to apply those values instead of a and b

(3x - 7)² = ( 3 x )² - 2 ( 3 x ) ( 7 ) + ( 7 )²

= 3²x² - 6 x ( 7 ) + 49

= 9 x² - 42 x + 49

Let us see the next example problem on "Expanding brackets with powers of 2".

Question 7 :

Expand (5x)² - 3 ²

Solution :

Here the question is in the form of (a²- b²) . Instead of a we have "5x" and instead of b we have "3" . So we need to apply the formula (a + b) (a - b) and we need to apply those values instead of a and b

(5x)² - 3² = (5x + 3) (5x - 3)

Let us see the next example problem on "Expanding brackets with powers of 2".

Question 8 :

Expand x² - 1 ²

Solution :

Here the question is in the form of (a²-b²) ². Instead of a we have "x" and instead of b we have "1" . So we need to apply the formula for square .That is (a+b)(a-b) and we need to apply those values instead of a and b

x² - 1² = (x + 1) (x-1)

Let us see the next example problem on "Expanding brackets with powers of 2".

Question 9 :

Expand 16 x² - 9

Solution :

We can split the first term 16 as 4x4 and 9 as 3x3. Instead of this we can write

= 4 ²x² - 3²

= (4x)² - 3 ²

= (4x + 3) (4x - 3)

Question 10 :

Expand (5x + 3y + 2z )²

Solution :

Here the question is in the form of (a + b + c) ². Instead of a we have "5x" instead of b we have "3y" and instead of c we have "2z". So we need to apply this formula.That is a²+b²+c²+2ab+2bc+2ca and we need to apply those values instead of a,b and c

= (5x)² + (3y)² + (2z)² + 2 (5x) (3y) + 2 (3y) (2z) + 2 (2z)(5x)

= 5²x² + 3²y² + 2²z² + 2 (15 x y) + 2 (6yz) + 2 (10zx)

= 25 x² + 9 y² + 4 z² + 30 x y + 12yz + 20 z x

Question 11 :

Expand (x - 2y + z )²

Solution :

Here the question is in the form of a plus b plus c whole square formula that is (a + b + c) ². Instead of a we have "x" instead of b we have "-2y" and instead of c we have "z" . So we need to apply the formula for squares.That is a² + b² + c² + 2 ab + 2 bc +2 ca and we need to apply those values instead of a,b and c

= (x)² + (-2y)² + (z)² + 2 (x) (-2y) + 2 (-2y) (z) + 2 (z) (x)

= x² + (-2)²y² + z² + 2 (x) (-2y) + 2(-2y) (z) + 2 (z)(x)

= x² + 4y² + z² -4 x y - 4 y z + 2 z x

## More Identities

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(a - b)² = a² - 2 ab + b²

a² - b² = (a + b) (a - b)

(x+a)(x+b)=x²+(a+b)x+ab

(a+b)³=a³+3a²b+3ab²+b³

(a-b)³=a³-3a²b+3ab²-b³

(a³+b³)= (a+b)(a²-ab+b²)

(a³-b³)=(a-b)(a²+ab+ b²)

(a+b+c)²= a²+b²+c²+2ab+2bc+2ca