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
Example: Express
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
5.If
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
Example: Find the square of
Example: Find the square of
![(2x-3y-4z)^2=[2x-(3y+4z)]^2=(2x)^2-2(2x)(3y+4z)+(3y+4z)^2](https://s0.wp.com/latex.php?latex=%282x-3y-4z%29%5E2%3D%5B2x-%283y%2B4z%29%5D%5E2%3D%282x%29%5E2-2%282x%29%283y%2B4z%29%2B%283y%2B4z%29%5E2+&bg=ffffff&fg=000&s=0&c=20201002)
![=4x^2-2(6xy+8xz)+[(3y)^2+2(3y)(4z)+(4z)^2]](https://s0.wp.com/latex.php?latex=%3D4x%5E2-2%286xy%2B8xz%29%2B%5B%283y%29%5E2%2B2%283y%29%284z%29%2B%284z%29%5E2%5D+&bg=ffffff&fg=000&s=0&c=20201002)
Example: Find the value of
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
ii)
4. If
5. If
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,
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
Example: Multiply
$latex
![(x^2+xy+y^2)( x^2-xy+y^2)=[(x^2+y^2)+xy][ (x^2+y^2)-xy]](https://s0.wp.com/latex.php?latex=%28x%5E2%2Bxy%2By%5E2%29%28+x%5E2-xy%2By%5E2%29%3D%5B%28x%5E2%2By%5E2%29%2Bxy%5D%5B+%28x%5E2%2By%5E2%29-xy%5D+&bg=ffffff&fg=000&s=0&c=20201002)
Example: Simplify:
![(a^2+ab+b^2)^2-(a^2-ab+b^2)^2=[(a^2+ab+b^2)+(a^2-ab+b^2)] \times [(a^2+ab+b^2)-(a^2-ab+b^2)]](https://s0.wp.com/latex.php?latex=%28a%5E2%2Bab%2Bb%5E2%29%5E2-%28a%5E2-ab%2Bb%5E2%29%5E2%3D%5B%28a%5E2%2Bab%2Bb%5E2%29%2B%28a%5E2-ab%2Bb%5E2%29%5D+%5Ctimes+%5B%28a%5E2%2Bab%2Bb%5E2%29-%28a%5E2-ab%2Bb%5E2%29%5D+&bg=ffffff&fg=000&s=0&c=20201002)
Example: Resolve into factors
Again,
Hence, the given expression becomes
Exercise:
1. Multiply together:
i)
ii)
[Hint: Take 200=(200+8) and 192=(200-8)]
iii)
iv)
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,
![(x-a)(x-b)=[x+(-a)][x+(-b)]=x^2+[(-a)+(-b)]x+[(-a) \times (-b)]](https://s0.wp.com/latex.php?latex=%28x-a%29%28x-b%29%3D%5Bx%2B%28-a%29%5D%5Bx%2B%28-b%29%5D%3Dx%5E2%2B%5B%28-a%29%2B%28-b%29%5Dx%2B%5B%28-a%29+%5Ctimes+%28-b%29%5D+&bg=ffffff&fg=000&s=0&c=20201002)
Similarly, the truth of the other two results can be proved.
Hence we can express the formula more clearly as follows:
