**Definition: **

Any general result expressed in symbols is called formula. In other words, a formula is the most general expression for any theorem respecting quantities.

**Formula: **

That is, the square of any two quantities is equal to the sum of their squares plus twice their product.

**Corrollary: **

**Example: Find the square of **

**Example: Simplify: **

**Example: Find the square of 8012.**

**Example: Find the value of when **

**Example: Express as a perfect square.**

**Exercise:**

1. Find the square of the following:

i)

ii)

2. Express each of the following expressions as a perfect square:

i)

ii)

3. Simplify:

i)

ii)

4. If , find the value of

5.If show that, and

## Formula:

That is, the square of the difference of any two quantities is equal to the sum of their squares minus twice their product.

**Corollary 1:**

**Corollary 2:** , and ,

and

**Example: Find the square of **

**Example: Find the square of **

**Example: Find the value of , when **

i.

ii.

iii.

**Exercise:**

1. Find the square of the following:

i)

ii) 993

[Hint: Write 993 as (1000-7)]

2. Express each of the following expressions as a perfect square:

i)

ii)

3. Simplify:

i)

[Hint: Put and ]

ii)

4. If , show that

5. If , show that

i)

ii)

ii)

## Formula:

That is, the product of the sum and the difference of any two quantities is equal to the difference of their squares.

Conversely, . Hence, we can always find the factors of an expression which is of the form

**Note:** When one expression is the product of two or more expressions, each of the latter is called a factor of the former.

**Example: Multiply by **

Example: Multiply by

$latex

**Example: Simplify: **

**Example: Resolve into factors **

Again,

Hence, the given expression becomes

## Exercise:

1. Multiply together:

i) and

ii) and

[Hint: Take 200=(200+8) and 192=(200-8)]

iii) and

iv) and

2. Simplify:

i)

ii)

Resolve into factors:

i)

ii)

iii)

iv)

**A few more formulae:**

Note: It is easy to notice that the above formula (3) includes the following results:

i.

ii.

iii.

For instance,

Similarly, the truth of the other two results can be proved.

Hence we can express the formula more clearly as follows:

## Leave a Reply