multiplication formulas

1. Multiplication Formulas

2. The most important product-formulas are
(The Squares) (A + B)(A + B) = (A + B) 2
(A – B)(A – B) = (A – B)2
(The Conjugates Product) (A + B)(A – B)
Multiplication Formulas

3. The most important product-formulas are
(The Squares) (A + B)(A + B) = (A + B) 2
(A – B)(A – B) = (A – B)2
(The Conjugates Product) (A + B)(A – B)
Multiplication Formulas
The Conjugates Product and the Difference of Squares

4. The most important product-formulas are
(The Squares) (A + B)(A + B) = (A + B) 2
(A – B)(A – B) = (A – B)2
(The Conjugates Product) (A + B)(A – B)
Multiplication Formulas
The two binomials (A + B) and (A – B) are said to be the
conjugate of each other.
The Conjugates Product and the Difference of Squares

5. The most important product-formulas are
(The Squares) (A + B)(A + B) = (A + B) 2
(A – B)(A – B) = (A – B)2
(The Conjugates Product) (A + B)(A – B)
Multiplication Formulas
The two binomials (A + B) and (A – B) are said to be the
conjugate of each other.
For example, the conjugate of (3x + 2) is (3x – 2),
and the conjugate of (2ab – c) is (2ab + c).
The Conjugates Product and the Difference of Squares

6. The most important product-formulas are
(The Squares) (A + B)(A + B) = (A + B) 2
(A – B)(A – B) = (A – B)2
(The Conjugates Product) (A + B)(A – B)
Multiplication Formulas
The two binomials (A + B) and (A – B) are said to be the
conjugate of each other.
The Conjugate Product:
(A + B)(A – B) = A2 – B2
For example, the conjugate of (3x + 2) is (3x – 2),
and the conjugate of (2ab – c) is (2ab + c).
The Conjugates Product and the Difference of Squares

7. The most important product-formulas are
(The Squares) (A + B)(A + B) = (A + B) 2
(A – B)(A – B) = (A – B)2
(The Conjugates Product) (A + B)(A – B)
Multiplication Formulas
The two binomials (A + B) and (A – B) are said to be the
conjugate of each other.
The Conjugate Product:
(A + B)(A – B) = A2 – B2
Verification: (A + B)(A – B) =
For example, the conjugate of (3x + 2) is (3x – 2),
and the conjugate of (2ab – c) is (2ab + c).
The Conjugates Product and the Difference of Squares

8. The most important product-formulas are
(The Squares) (A + B)(A + B) = (A + B) 2
(A – B)(A – B) = (A – B)2
(The Conjugates Product) (A + B)(A – B)
Multiplication Formulas
The two binomials (A + B) and (A – B) are said to be the
conjugate of each other.
The Conjugate Product:
(A + B)(A – B) = A2 – B2
Verification: (A + B)(A – B) = A2 – AB + AB – B2
For example, the conjugate of (3x + 2) is (3x – 2),
and the conjugate of (2ab – c) is (2ab + c).
The Conjugates Product and the Difference of Squares

9. The most important product-formulas are
(The Squares) (A + B)(A + B) = (A + B) 2
(A – B)(A – B) = (A – B)2
(The Conjugates Product) (A + B)(A – B)
Multiplication Formulas
The two binomials (A + B) and (A – B) are said to be the
conjugate of each other.
The Conjugate Product:
(A + B)(A – B) = A2 – B2
Verification: (A + B)(A – B) = A2 – AB + AB – B2 = A2 – B2
For example, the conjugate of (3x + 2) is (3x – 2),
and the conjugate of (2ab – c) is (2ab + c).
The Conjugates Product and the Difference of Squares

10. The most important product-formulas are
(The Squares) (A + B)(A + B) = (A + B) 2
(A – B)(A – B) = (A – B)2
(The Conjugates Product) (A + B)(A – B)
Multiplication Formulas
The two binomials (A + B) and (A – B) are said to be the
conjugate of each other.
The Conjugate Product:
(A + B)(A – B) = A2 – B2
Verification: (A + B)(A – B) = A2 – AB + AB – B2 = A2 – B2
Conjugate Product Difference of Squares
For example, the conjugate of (3x + 2) is (3x – 2),
and the conjugate of (2ab – c) is (2ab + c).
The Conjugates Product and the Difference of Squares

11. The most important product-formulas are
(The Squares) (A + B)(A + B) = (A + B) 2
(A – B)(A – B) = (A – B)2
(The Conjugates Product) (A + B)(A – B)
Multiplication Formulas
The two binomials (A + B) and (A – B) are said to be the
conjugate of each other.
The Conjugate Product:
(A + B)(A – B) = A2 – B2
Verification: (A + B)(A – B) = A2 – AB + AB – B2 = A2 – B2
For example, the conjugate of (3x + 2) is (3x – 2),
and the conjugate of (2ab – c) is (2ab + c).
The Conjugates Product and the Difference of Squares
Note: The conjugate of (3x + 2) or (3x – 2) is different from
the opposite of (3x + 2) which is (–3x – 2).
Conjugate Product Difference of Squares

12. Multiplication Formulas
Here are some examples of squaring: (3x)2 =

13. Multiplication Formulas
Here are some examples of squaring: (3x)2 = 9x2,

14. Multiplication Formulas
Here are some examples of squaring: (3x)2 = 9x2,
(2xy)2 =

15. Multiplication Formulas
Here are some examples of squaring: (3x)2 = 9x2,
(2xy)2 = 4x2y2,

16. Multiplication Formulas
Here are some examples of squaring: (3x)2 = 9x2,
(2xy)2 = 4x2y2, and (5z2)2

17. Multiplication Formulas
Here are some examples of squaring: (3x)2 = 9x2,
(2xy)2 = 4x2y2, and (5z2)2 = 25z4.

18. Multiplication Formulas
Example A. Expand using the formula.
a. (3x + 2)(3x – 2)
Here are some examples of squaring: (3x)2 = 9x2,
(2xy)2 = 4x2y2, and (5z2)2 = 25z4.

19. Multiplication Formulas
Example A. Expand using the formula.
a. (3x + 2)(3x – 2)
(A + B)(A – B)
Here are some examples of squaring: (3x)2 = 9x2,
(2xy)2 = 4x2y2, and (5z2)2 = 25z4.

20. Multiplication Formulas
Example A. Expand using the formula.
a. (3x + 2)(3x – 2) = (3x)2 – (2)2
(A + B)(A – B) = A2 – B2
Here are some examples of squaring: (3x)2 = 9x2,
(2xy)2 = 4x2y2, and (5z2)2 = 25z4.

21. Multiplication Formulas
Example A. Expand using the formula.
a. (3x + 2)(3x – 2) = (3x)2 – (2)2 = 9x2 – 4
(A + B)(A – B) = A2 – B2
Here are some examples of squaring: (3x)2 = 9x2,
(2xy)2 = 4x2y2, and (5z2)2 = 25z4.

22. Multiplication Formulas
Example A. Expand using the formula.
a. (3x + 2)(3x – 2) = (3x)2 – (2)2 = 9x2 – 4
(A + B)(A – B) = A2 – B2
b. (2xy – 5z2)(2xy + 5z2)
Here are some examples of squaring: (3x)2 = 9x2,
(2xy)2 = 4x2y2, and (5z2)2 = 25z4.

23. Multiplication Formulas
Example A. Expand using the formula.
a. (3x + 2)(3x – 2) = (3x)2 – (2)2 = 9x2 – 4
(A + B)(A – B) = A2 – B2
b. (2xy – 5z2)(2xy + 5z2)
= (2xy)2 – (5z2)2
Here are some examples of squaring: (3x)2 = 9x2,
(2xy)2 = 4x2y2, and (5z2)2 = 25z4.

24. Multiplication Formulas
Example A. Expand using the formula.
a. (3x + 2)(3x – 2) = (3x)2 – (2)2 = 9x2 – 4
(A + B)(A – B) = A2 – B2
b. (2xy – 5z2)(2xy + 5z2)
= (2xy)2 – (5z2)2
= 4x2y2 – 25z4
Here are some examples of squaring: (3x)2 = 9x2,
(2xy)2 = 4x2y2, and (5z2)2 = 25z4.

25. Multiplication Formulas
Example A. Expand using the formula.
a. (3x + 2)(3x – 2) = (3x)2 – (2)2 = 9x2 – 4
(A + B)(A – B) = A2 – B2
b. (2xy – 5z2)(2xy + 5z2)
= (2xy)2 – (5z2)2
= 4x2y2 – 25z4
Here are some examples of squaring: (3x)2 = 9x2,
(2xy)2 = 4x2y2, and (5z2)2 = 25z4.
II. Square Formulas

26. Multiplication Formulas
Example A. Expand using the formula.
a. (3x + 2)(3x – 2) = (3x)2 – (2)2 = 9x2 – 4
(A + B)(A – B) = A2 – B2
b. (2xy – 5z2)(2xy + 5z2)
= (2xy)2 – (5z2)2
= 4x2y2 – 25z
Here are some examples of squaring: (3x)2 = 9x2,
(2xy)2 = 4x2y2, and (5z2)2 = 25z4.
II. Square Formulas

27. Multiplication Formulas
Example A. Expand using the formula.
a. (3x + 2)(3x – 2) = (3x)2 – (2)2 = 9x2 – 4
(A + B)(A – B) = A2 – B2
b. (2xy – 5z2)(2xy + 5z2)
= (2xy)2 – (5z2)2
= 4x2y2 – 25z
Here are some examples of squaring: (3x)2 = 9x2,
(2xy)2 = 4x2y2, and (5z2)2 = 25z4.
II. Square Formulas
(A + B)2 = A2 + 2AB + B2
(A – B)2 = A2 – 2AB + B2

28. Multiplication Formulas
Example A. Expand using the formula.
a. (3x + 2)(3x – 2) = (3x)2 – (2)2 = 9x2 – 4
(A + B)(A – B) = A2 – B2
b. (2xy – 5z2)(2xy + 5z2)
= (2xy)2 – (5z2)2
= 4x2y2 – 25z
Here are some examples of squaring: (3x)2 = 9x2,
(2xy)2 = 4x2y2, and (5z2)2 = 25z4.
II. Square Formulas
(A + B)2 = A2 + 2AB + B2
(A – B)2 = A2 – 2AB + B2
Check this by multiplying,
(A + B)2 = (A + B)(A + B)

29. Multiplication Formulas
Example A. Expand using the formula.
a. (3x + 2)(3x – 2) = (3x)2 – (2)2 = 9x2 – 4
(A + B)(A – B) = A2 – B2
b. (2xy – 5z2)(2xy + 5z2)
= (2xy)2 – (5z2)2
= 4x2y2 – 25z
Here are some examples of squaring: (3x)2 = 9x2,
(2xy)2 = 4x2y2, and (5z2)2 = 25z4.
II. Square Formulas
(A + B)2 = A2 + 2AB + B2
(A – B)2 = A2 – 2AB + B2
Check this by multiplying,
(A + B)2 = (A + B)(A + B) = A2 + AB + BA + B2

30. Multiplication Formulas
Example A. Expand using the formula.
a. (3x + 2)(3x – 2) = (3x)2 – (2)2 = 9x2 – 4
(A + B)(A – B) = A2 – B2
b. (2xy – 5z2)(2xy + 5z2)
= (2xy)2 – (5z2)2
= 4x2y2 – 25z
Here are some examples of squaring: (3x)2 = 9x2,
(2xy)2 = 4x2y2, and (5z2)2 = 25z4.
II. Square Formulas
(A + B)2 = A2 + 2AB + B2
(A – B)2 = A2 – 2AB + B2
Check this by multiplying,
(A + B)2 = (A + B)(A + B) = A2 + AB + BA + B2 = A2 + 2AB + B2
When squaring, we obtain
two identical copies of AB.
For conjugates, we obtain
AB and –AB which cancels.

31. Multiplication Formulas
Example A. Expand using the formula.
a. (3x + 2)(3x – 2) = (3x)2 – (2)2 = 9x2 – 4
(A + B)(A – B) = A2 – B2
b. (2xy – 5z2)(2xy + 5z2)
= (2xy)2 – (5z2)2
= 4x2y2 – 25z
Here are some examples of squaring: (3x)2 = 9x2,
(2xy)2 = 4x2y2, and (5z2)2 = 25z4.
II. Square Formulas
(A + B)2 = A2 + 2AB + B2
(A – B)2 = A2 – 2AB + B2
Check this by multiplying,
(A + B)2 = (A + B)(A + B) = A2 + AB + BA + B2 = A2 + 2AB + B2
We say that “(A + B)2 is A2, B2, plus twice A*B”,
and “(A – B)2 is A2, B2, minus twice A*B”.
When squaring, we obtain
two identical copies of AB.
For conjugates, we obtain
AB and –AB which cancels.

32. Example B. Expand using the formula.
a. (3x + 4)2
Multiplication Formulas

33. Example B. Expand using the formula.
a. (3x + 4)2
(A + B)2
Multiplication Formulas

34. Example B. Expand using the formula.
a. (3x + 4)2
(A + B)2 = A2 + 2AB + B2
Multiplication Formulas

35. Example B. Expand using the formula.
a. (3x + 4)2 = (3x)2
(A + B)2 = A2 + 2AB + B2
Multiplication Formulas

36. Example B. Expand using the formula.
a. (3x + 4)2 = (3x)2 + 2(3x)(4)
(A + B)2 = A2 + 2AB + B2
Multiplication Formulas

37. Example B. Expand using the formula.
a. (3x + 4)2 = (3x)2 + 2(3x)(4) + 42
(A + B)2 = A2 + 2AB + B2
Multiplication Formulas

38. Example B. Expand using the formula.
a. (3x + 4)2 = (3x)2 + 2(3x)(4) + 42 = 9x2 + 24x + 16
(A + B)2 = A2 + 2AB + B2
Multiplication Formulas

39. Example B. Expand using the formula.
a. (3x + 4)2 = (3x)2 + 2(3x)(4) + 42 = 9x2 + 24x + 16
(A + B)2 = A2 + 2AB + B2
Multiplication Formulas
b. (3a – 5b)2

40. Example B. Expand using the formula.
a. (3x + 4)2 = (3x)2 + 2(3x)(4) + 42 = 9x2 + 24x + 16
(A + B)2 = A2 + 2AB + B2
Multiplication Formulas
b. (3a – 5b)2 = (3a)2 – 2(3a)(5b) + (5b)2

41. Example B. Expand using the formula.
a. (3x + 4)2 = (3x)2 + 2(3x)(4) + 42 = 9x2 + 24x + 16
(A + B)2 = A2 + 2AB + B2
Multiplication Formulas
b. (3a – 5b)2 = (3a)2 – 2(3a)(5b) + (5b)2
= 9a2 – 30ab + 25b2

42. Example B. Expand using the formula.
a. (3x + 4)2 = (3x)2 + 2(3x)(4) + 42 = 9x2 + 24x + 16
(A + B)2 = A2 + 2AB + B2
Multiplication Formulas
b. (3a – 5b)2 = (3a)2 – 2(3a)(5b) + (5b)2
= 9a2 – 30ab + 25b2
III. Some Applications of the Formulas

43. Example B. Expand using the formula.
a. (3x + 4)2 = (3x)2 + 2(3x)(4) + 42 = 9x2 + 24x + 16
(A + B)2 = A2 + 2AB + B2
Multiplication Formulas
b. (3a – 5b)2 = (3a)2 – 2(3a)(5b) + (5b)2
= 9a2 – 30ab + 25b2
III. Some Applications of the Formulas
We can use the above formulas to help us multiply numbers.

44. Example B. Expand using the formula.
a. (3x + 4)2 = (3x)2 + 2(3x)(4) + 42 = 9x2 + 24x + 16
(A + B)2 = A2 + 2AB + B2
Multiplication Formulas
b. (3a – 5b)2 = (3a)2 – 2(3a)(5b) + (5b)2
= 9a2 – 30ab + 25b2
III. Some Applications of the Formulas
We can use the above formulas to help us multiply numbers.
Example C. Calculate. Use the conjugate formula.
a. 51*49

45. Example B. Expand using the formula.
a. (3x + 4)2 = (3x)2 + 2(3x)(4) + 42 = 9x2 + 24x + 16
(A + B)2 = A2 + 2AB + B2
Multiplication Formulas
b. (3a – 5b)2 = (3a)2 – 2(3a)(5b) + (5b)2
= 9a2 – 30ab + 25b2
III. Some Applications of the Formulas
We can use the above formulas to help us multiply numbers.
Example C. Calculate. Use the conjugate formula.
a. 51*49 = (50 + 1)(50 – 1)

46. Example B. Expand using the formula.
a. (3x + 4)2 = (3x)2 + 2(3x)(4) + 42 = 9x2 + 24x + 16
(A + B)2 = A2 + 2AB + B2
Multiplication Formulas
b. (3a – 5b)2 = (3a)2 – 2(3a)(5b) + (5b)2
= 9a2 – 30ab + 25b2
III. Some Applications of the Formulas
We can use the above formulas to help us multiply numbers.
Example C. Calculate. Use the conjugate formula.
a. 51*49 = (50 + 1)(50 – 1) = 502 – 12

47. Example B. Expand using the formula.
a. (3x + 4)2 = (3x)2 + 2(3x)(4) + 42 = 9x2 + 24x + 16
(A + B)2 = A2 + 2AB + B2
Multiplication Formulas
b. (3a – 5b)2 = (3a)2 – 2(3a)(5b) + (5b)2
= 9a2 – 30ab + 25b2
III. Some Applications of the Formulas
We can use the above formulas to help us multiply numbers.
Example C. Calculate. Use the conjugate formula.
a. 51*49 = (50 + 1)(50 – 1) = 502 – 12 = 2,500 – 1 = 2,499

48. Example B. Expand using the formula.
a. (3x + 4)2 = (3x)2 + 2(3x)(4) + 42 = 9x2 + 24x + 16
(A + B)2 = A2 + 2AB + B2
Multiplication Formulas
b. (3a – 5b)2 = (3a)2 – 2(3a)(5b) + (5b)2
= 9a2 – 30ab + 25b2
III. Some Applications of the Formulas
We can use the above formulas to help us multiply numbers.
Example C. Calculate. Use the conjugate formula.
a. 51*49 = (50 + 1)(50 – 1) = 502 – 12 = 2,500 – 1 = 2,499
b. 52*48

49. Example B. Expand using the formula.
a. (3x + 4)2 = (3x)2 + 2(3x)(4) + 42 = 9x2 + 24x + 16
(A + B)2 = A2 + 2AB + B2
Multiplication Formulas
b. (3a – 5b)2 = (3a)2 – 2(3a)(5b) + (5b)2
= 9a2 – 30ab + 25b2
III. Some Applications of the Formulas
We can use the above formulas to help us multiply numbers.
Example C. Calculate. Use the conjugate formula.
a. 51*49 = (50 + 1)(50 – 1) = 502 – 12 = 2,500 – 1 = 2,499
b. 52*48 = (50 + 2)(50 – 2) = 502 – 22

50. Example B. Expand using the formula.
a. (3x + 4)2 = (3x)2 + 2(3x)(4) + 42 = 9x2 + 24x + 16
(A + B)2 = A2 + 2AB + B2
Multiplication Formulas
b. (3a – 5b)2 = (3a)2 – 2(3a)(5b) + (5b)2
= 9a2 – 30ab + 25b2
III. Some Applications of the Formulas
We can use the above formulas to help us multiply numbers.
Example C. Calculate. Use the conjugate formula.
a. 51*49 = (50 + 1)(50 – 1) = 502 – 12 = 2,500 – 1 = 2,499
b. 52*48 = (50 + 2)(50 – 2) = 502 – 22 = 2,500 – 4 = 2,496

51. Example B. Expand using the formula.
a. (3x + 4)2 = (3x)2 + 2(3x)(4) + 42 = 9x2 + 24x + 16
(A + B)2 = A2 + 2AB + B2
Multiplication Formulas
b. (3a – 5b)2 = (3a)2 – 2(3a)(5b) + (5b)2
= 9a2 – 30ab + 25b2
III. Some Applications of the Formulas
We can use the above formulas to help us multiply numbers.
Example C. Calculate. Use the conjugate formula.
a. 51*49 = (50 + 1)(50 – 1) = 502 – 12 = 2,500 – 1 = 2,499
b. 52*48 = (50 + 2)(50 – 2) = 502 – 22 = 2,500 – 4 = 2,496
c. 63*57 =

52. Example B. Expand using the formula.
a. (3x + 4)2 = (3x)2 + 2(3x)(4) + 42 = 9x2 + 24x + 16
(A + B)2 = A2 + 2AB + B2
Multiplication Formulas
b. (3a – 5b)2 = (3a)2 – 2(3a)(5b) + (5b)2
= 9a2 – 30ab + 25b2
III. Some Applications of the Formulas
We can use the above formulas to help us multiply numbers.
Example C. Calculate. Use the conjugate formula.
a. 51*49 = (50 + 1)(50 – 1) = 502 – 12 = 2,500 – 1 = 2,499
b. 52*48 = (50 + 2)(50 – 2) = 502 – 22 = 2,500 – 4 = 2,496
c. 63*57 = (60 + 3)(60 – 3) = 602 – 32

53. Example B. Expand using the formula.
a. (3x + 4)2 = (3x)2 + 2(3x)(4) + 42 = 9x2 + 24x + 16
(A + B)2 = A2 + 2AB + B2
Multiplication Formulas
b. (3a – 5b)2 = (3a)2 – 2(3a)(5b) + (5b)2
= 9a2 – 30ab + 25b2
III. Some Applications of the Formulas
We can use the above formulas to help us multiply numbers.
Example C. Calculate. Use the conjugate formula.
a. 51*49 = (50 + 1)(50 – 1) = 502 – 12 = 2,500 – 1 = 2,499
b. 52*48 = (50 + 2)(50 – 2) = 502 – 22 = 2,500 – 4 = 2,496
c. 63*57 = (60 + 3)(60 – 3) = 602 – 32 = 3,600 – 9 = 3,591

54. Example B. Expand using the formula.
a. (3x + 4)2 = (3x)2 + 2(3x)(4) + 42 = 9x2 + 24x + 16
(A + B)2 = A2 + 2AB + B2
Multiplication Formulas
b. (3a – 5b)2 = (3a)2 – 2(3a)(5b) + (5b)2
= 9a2 – 30ab + 25b2
III. Some Applications of the Formulas
We can use the above formulas to help us multiply numbers.
The conjugate formula
(A + B)(A – B) = A2 – B2
may be used to multiply two numbers of the forms
(A + B) and (A – B) where A2 and B2 can be calculated easily.
Example C. Calculate. Use the conjugate formula.
a. 51*49 = (50 + 1)(50 – 1) = 502 – 12 = 2,500 – 1 = 2,499
b. 52*48 = (50 + 2)(50 – 2) = 502 – 22 = 2,500 – 4 = 2,496
c. 63*57 = (60 + 3)(60 – 3) = 602 – 32 = 3,600 – 9 = 3,591

55. Multiplication Formulas
“(A + B)2 is A2, B2, plus twice A*B”,
“(A – B)2 is A2, B2, minus twice A*B”.
The Squaring Formulas.

56. Multiplication Formulas
Example D. Calculate. Use the squaring formulas.
a. 512
“(A + B)2 is A2, B2, plus twice A*B”,
“(A – B)2 is A2, B2, minus twice A*B”.
The Squaring Formulas.

57. Multiplication Formulas
Example D. Calculate. Use the squaring formulas.
a. 512 = (50 + 1)2
“(A + B)2 is A2, B2, plus twice A*B”,
“(A – B)2 is A2, B2, minus twice A*B”.
The Squaring Formulas.

58. Multiplication Formulas
Example D. Calculate. Use the squaring formulas.
a. 512 = (50 + 1)2 = 502 + 12
“(A + B)2 is A2, B2, plus twice A*B”,
“(A – B)2 is A2, B2, minus twice A*B”.
The Squaring Formulas.

59. Multiplication Formulas
Example D. Calculate. Use the squaring formulas.
a. 512 = (50 + 1)2 = 502 + 12 + 2(50)(1)
“(A + B)2 is A2, B2, plus twice A*B”,
“(A – B)2 is A2, B2, minus twice A*B”.
The Squaring Formulas.

60. Multiplication Formulas
Example D. Calculate. Use the squaring formulas.
a. 512 = (50 + 1)2 = 502 + 12 + 2(50)(1)
= 2,500 + 1 + 100
“(A + B)2 is A2, B2, plus twice A*B”,
“(A – B)2 is A2, B2, minus twice A*B”.
The Squaring Formulas.

61. Multiplication Formulas
Example D. Calculate. Use the squaring formulas.
a. 512 = (50 + 1)2 = 502 + 12 + 2(50)(1)
= 2,500 + 1 + 100
= 2,601
“(A + B)2 is A2, B2, plus twice A*B”,
“(A – B)2 is A2, B2, minus twice A*B”.
The Squaring Formulas.

62. Multiplication Formulas
Example D. Calculate. Use the squaring formulas.
a. 512 = (50 + 1)2 = 502 + 12 + 2(50)(1)
= 2,500 + 1 + 100
= 2,601
“(A + B)2 is A2, B2, plus twice A*B”,
“(A – B)2 is A2, B2, minus twice A*B”.
The Squaring Formulas.
b. 492

63. Multiplication Formulas
Example D. Calculate. Use the squaring formulas.
a. 512 = (50 + 1)2 = 502 + 12 + 2(50)(1)
= 2,500 + 1 + 100
= 2,601
“(A + B)2 is A2, B2, plus twice A*B”,
“(A – B)2 is A2, B2, minus twice A*B”.
The Squaring Formulas.
b. 492 = (50 – 1)2

64. Multiplication Formulas
Example D. Calculate. Use the squaring formulas.
a. 512 = (50 + 1)2 = 502 + 12 + 2(50)(1)
= 2,500 + 1 + 100
= 2,601
“(A + B)2 is A2, B2, plus twice A*B”,
“(A – B)2 is A2, B2, minus twice A*B”.
The Squaring Formulas.
b. 492 = (50 – 1)2 = 502 + 12

65. Multiplication Formulas
Example D. Calculate. Use the squaring formulas.
a. 512 = (50 + 1)2 = 502 + 12 + 2(50)(1)
= 2,500 + 1 + 100
= 2,601
“(A + B)2 is A2, B2, plus twice A*B”,
“(A – B)2 is A2, B2, minus twice A*B”.
The Squaring Formulas.
b. 492 = (50 – 1)2 = 502 + 12 – 2(50)(1)

66. Multiplication Formulas
Example D. Calculate. Use the squaring formulas.
a. 512 = (50 + 1)2 = 502 + 12 + 2(50)(1)
= 2,500 + 1 + 100
= 2,601
“(A + B)2 is A2, B2, plus twice A*B”,
“(A – B)2 is A2, B2, minus twice A*B”.
The Squaring Formulas.
b. 492 = (50 – 1)2 = 502 + 12 – 2(50)(1)
= 2,500 + 1 – 100

67. Multiplication Formulas
Example D. Calculate. Use the squaring formulas.
a. 512 = (50 + 1)2 = 502 + 12 + 2(50)(1)
= 2,500 + 1 + 100
= 2,601
“(A + B)2 is A2, B2, plus twice A*B”,
“(A – B)2 is A2, B2, minus twice A*B”.
The Squaring Formulas.
b. 492 = (50 – 1)2 = 502 + 12 – 2(50)(1)
= 2,500 + 1 – 100
= 2,401

68. Multiplication Formulas
Example D. Calculate. Use the squaring formulas.
a. 512 = (50 + 1)2 = 502 + 12 + 2(50)(1)
= 2,500 + 1 + 100
= 2,601
“(A + B)2 is A2, B2, plus twice A*B”,
“(A – B)2 is A2, B2, minus twice A*B”.
The Squaring Formulas.
b. 492 = (50 – 1)2 = 502 + 12 – 2(50)(1)
= 2,500 + 1 – 100
= 2,401
b. (50½) 2 = (50 + ½ )2

69. Multiplication Formulas
Example D. Calculate. Use the squaring formulas.
a. 512 = (50 + 1)2 = 502 + 12 + 2(50)(1)
= 2,500 + 1 + 100
= 2,601
“(A + B)2 is A2, B2, plus twice A*B”,
“(A – B)2 is A2, B2, minus twice A*B”.
The Squaring Formulas.
b. 492 = (50 – 1)2 = 502 + 12 – 2(50)(1)
= 2,500 + 1 – 100
= 2,401
b. (50½) 2 = (50 + ½ )2 = 502 + ½ 2

70. Multiplication Formulas
Example D. Calculate. Use the squaring formulas.
a. 512 = (50 + 1)2 = 502 + 12 + 2(50)(1)
= 2,500 + 1 + 100
= 2,601
“(A + B)2 is A2, B2, plus twice A*B”,
“(A – B)2 is A2, B2, minus twice A*B”.
The Squaring Formulas.
b. 492 = (50 – 1)2 = 502 + 12 – 2(50)(1)
= 2,500 + 1 – 100
= 2,401
b. (50½) 2 = (50 + ½ )2 = 502 + ½ 2 + 2 (½) (50)

71. Multiplication Formulas
Example D. Calculate. Use the squaring formulas.
a. 512 = (50 + 1)2 = 502 + 12 + 2(50)(1)
= 2,500 + 1 + 100
= 2,601
“(A + B)2 is A2, B2, plus twice A*B”,
“(A – B)2 is A2, B2, minus twice A*B”.
The Squaring Formulas.
b. 492 = (50 – 1)2 = 502 + 12 – 2(50)(1)
= 2,500 + 1 – 100
= 2,401
b. (50½) 2 = (50 + ½ )2 = 502 + ½ 2 + 2 (½) (50)
= 2,500 + 1/4 + 50

72. Multiplication Formulas
Example D. Calculate. Use the squaring formulas.
a. 512 = (50 + 1)2 = 502 + 12 + 2(50)(1)
= 2,500 + 1 + 100
= 2,601
“(A + B)2 is A2, B2, plus twice A*B”,
“(A – B)2 is A2, B2, minus twice A*B”.
The Squaring Formulas.
b. 492 = (50 – 1)2 = 502 + 12 – 2(50)(1)
= 2,500 + 1 – 100
= 2,401
b. (50½) 2 = (50 + ½ )2 = 502 + ½ 2 + 2 (½) (50)
= 2,500 + 1/4 + 50
= 2,550¼
We may take advantage of these formulas tor
lengthier multiplication.

73. Example E. Expand using the formula.
(3x – 2y + 4) (3x – 2y – 4)
Multiplication Formulas

74. Example E. Expand using the formula.
(3x – 2y + 4) (3x – 2y – 4)
= [(3x – 2y) + 4] [(3x – 2y) – 4]
Multiplication Formulas
group into conjugates

75. Example E. Expand using the formula.
(3x – 2y + 4) (3x – 2y – 4)
= [(3x – 2y) + 4] [(3x – 2y) – 4]
= (3x – 2y)2 – 42
Multiplication Formulas
group into conjugates
difference of squares

76. Example E. Expand using the formula.
(3x – 2y + 4) (3x – 2y – 4)
= [(3x – 2y) + 4] [(3x – 2y) – 4]
= (3x – 2y)2 – 42
= (3x)2 – 2(3x)(2y) + (2y)2 – 42
= 9x2 + 12xy + 4y2 – 16
Multiplication Formulas
group into conjugates
difference of squares

77. Exercise. A. Calculate. Use the conjugate formula.
Multiplication Formulas
1. 21*19 2. 31*29 3. 41*39 4. 71*69
5. 22*18 6. 32*28 7. 52*48 8. 73*67
B. Calculate. Use the squaring formula.
9. 212 10. 312 11. 392 12. 692
13. 982 14. 30½2 15. 100½2 16. 49½2
18. (x + 5)(x – 5) 19. (x – 7)(x + 7)
20. (2x + 3)(2x – 3) 21. (3x – 5)(3x + 5)
C. Expand.
22. (7x + 2)(7x – 2) 23. (–7 + 3x )(–7 – 3x)
24. (–4x + 3)(–4x – 3) 25. (2x – 3y)(2x + 3y)
26. (4x – 5y)(5x + 5y) 27. (1 – 7y)(1 + 7y)
28. (5 – 3x)(5 + 3x) 29. (10 + 9x)(10 – 9x)
30. (x + 5)2 31. (x – 7)2
32. (2x + 3)2 33. (3x + 5y)2
34. (7x – 2y)2 35. (2x – h)2