# What values of b satisfy 3(2b + 3)2 = 36?

This question aims to find the values of b from the given equation by using arithmetic laws. The simple use of addition and multiplication with values within the brackets will give the value of b.

Arithmetic is the oldest branch of mathematics and the word arithmetic originated from the Greek word “Arithmos,” meaning number. This branch of mathematics deals with  basic operations like addition, multiplication, division, and subtraction. It is the in-depth study of the laws and properties of these operations.

To solve these equations, we need to follow some order of applying operations. The order of operation is applying brackets first, then the operation of division. After division, apply multiplication and then addition and subtraction.

From the given equation:

$3 ( 2b + 3 ) ^ { 2 } = 36$

$( 2b + 3 ) ^ { 2 } = \frac { 36 }{ 3 }$

$( 2b + 3 ) ^ { 2 } = 12$

Taking square root on both sides:

$2b + 3 = \pm \sqrt { 12 }$

$2b = \pm \sqrt { 12 } – 3$

Dividing the equation by 2:

$b = \frac { \pm 2\sqrt { 3 } – 3 } {2}$

$b = \frac { – 3 + 2\sqrt { 3 }} {2}$

$b = \frac { -3 – 2\sqrt { 3 }} {2}$

## Numerical Results

The values of b are $b = \frac { – 3 + 2\sqrt { 3 }} {2}$ and $b = \frac { -3 – 2\sqrt { 3 }} {2}$.

## Example

Find the value of b if the equation is $3 ( 4b + 3 ) ^ {2} = 9$

From the given equation:

$3 ( 4b + 3 ) ^ {2} = 9$

$( 4b + 3 ) ^ {2} = \frac { 9 }{ 3 }$

$( 4b + 3 ) ^ {2} = 3$

Taking the square root on both sides:

$4b + 3 = \pm \sqrt { 3 }$

$4b = \pm \sqrt { 3 } – 3$

Dividing the equation by 4:

$b = \frac { \pm \sqrt 3 – 3 } { 4 }$

By rearranging the equation:

$b = \frac { – 3 + \sqrt 3 } { 4 }$

$b = \frac { -3 – \sqrt 3 } { 2 }$

For a simple equation:

$2 ( 5b + 3 ) = 10$

$10b + 6 = 10$

$10b = 10 – 6$

$10b = 4$

$b = \frac { 4 } { 10 }$

$b = \frac { 2 } { 5 }$

The value of b is $b = \frac { 2 } { 5 }$.

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