- ALGEBRA –
FORMULAE OF EXPANSIONS -
1) (a + b)² = a² + 2ab + b²
2) (a – b)² = a² - 2ab + b²
3) (a + b)² + (a – b)² = 2 (a² + b²)
4) (a + b)² - (a – b)² = 4ab
5) (a + b) (a – b) = a² - b²
1 1
6) (a + ----- )² = a² + ------ + 2
a a²
1 1
7) (a - ------ )² = a² + ------ - 2
a a²
1 1 1
8) (a + ------ ) (a - ------ ) = a² - ------
a a a²
1 1 1
9) (a + ------ )² + (a - ------ )² = 2 (a² + ------ )
a a a²
1 1
10) (a + ------ )² - (a - ------- )² = 4
a a
11) ( a + b + c )² = a² + b² + c² + 2 (ab + bc + ca)
12) (a) (x + a) (x + b) = x² + (a + b) x + ab
(b) (x + a) (x - b) = x² + (a – b) x – ab
(c) (x – a) (x + b ) = x² - (a – b) x – ab
(d) (x – a) (x – b) = x² - (a + b) x + ab
Some more product –
13) (x + a) (x + b) = (x + a) x + (x + a) b
= x² + ax + bx + ab
= x² + (a + b)x + ab
So, (x + a) (x + b)
= x² + (algebraic sum of 2nd terms) x + (product of 2nd terms)
14) (x – a) (x – b) = (x – a) x – (x – a) b
= x² - ax – bx + ab
= x² - (a + b)x + ab
So, (x - a) (x - b)
= x² - (algebraic sum of 2nd terms) x + (product of 2nd terms)
15) (x + a) (x – b) = (x + a) x – (x + a) b
= x² + ax – bx – ab
= x² + (a – b)x – ab
So, (x + a) (x - b) = x² + (algebraic subtraction of 2nd terms) x - (product of 2nd terms)
16) (x – a) (x + b) = (x – a) x + (x – a) b
= x² - ax + bx – ab
= x² - (a – b)x - ab
So, (x - a) (x + b)
= x² - (algebraic subtraction of 2nd terms) x - (product of 2nd terms)
To find the product (ax + by) (cx + dy)
Method –
Step.1) Multiply the first term
Step.2) Find the inner and outer products and take their algebraic sums
Step.3) Multiply the last terms,
Thus, we have –
(ax + by) (cx + dy) = (ax) (cx + dy) + (by) (cx + dy)
= (ax) (cx) + (ax) (dy) + (by) (cx) + (by) (dy)
= (ax) (cx) + {(by) (cx) + (ax) (dy)} + (by) (dy)
Here, (by) (cx) is inner product and (ax) (dy) is the outer product.