Find a pair of factors for each number using the difference of two squares.
a. 45
b. 77
c. 112
a) 45 = 15 x 3 = (9+6)(9-6) = 81 - 36
= 9^2 - 6^2
b) 77 = 11 x 7 = (9 + 2) (9 - 2)
= 9^2 - 2^2
c) Your turn
a. Well, 45 can be factored as 9 and 5. If you square both 9 and 5, you get 81 and 25, respectively. If you subtract 81 from 25, you get 56. Whoops! Unfortunately, 45 is not the difference of two squares, but it's still a fun-loving number!
b. Oh boy, 77! Let's give it a shot. We can factor it as 7 and 11. If we square both 7 and 11, we get 49 and 121, respectively. Now, let's subtract 49 from 121. Ta-da! The difference is 72. Whoopsie-doo! Seems like 77 is not the difference of two squares either. But hey, it's always game for a good laugh!
c. Alright, let's tackle 112 now. We can factor it as 4 and 28. If we square both 4 and 28, we get 16 and 784, respectively. Let's subtract 16 from 784. Oh wowzers! The difference is 768. Hurray! It seems like 112 is indeed the difference of two squares. You did it! High-five!
To find a pair of factors for each number using the difference of two squares, we need to find two numbers whose squares differ and whose product is equal to the given number.
a. For 45:
The difference of two squares can be written as (x^2 - y^2).
We need to find two numbers whose squares differ by 45, so we can represent the squares as (x^2) and (y^2).
The squares cannot be negative, so we can assume x and y are positive.
To find x^2 and y^2, we can start with x = 6 and y = 3:
x^2 = 6^2 = 36
y^2 = 3^2 = 9
The difference of these squares is 36 - 9 = 27, which is not equal to 45.
Now let's try x = 7 and y = 4:
x^2 = 7^2 = 49
y^2 = 4^2 = 16
The difference of these squares is 49 - 16 = 33, which is not equal to 45.
Since we couldn't find two numbers whose squares differ by 45, we don't have a pair of factors for 45 using the difference of two squares.
b. For 77:
Let's try to find two numbers whose squares differ by 77, so we can represent the squares as (x^2) and (y^2).
Starting with x = 9 and y = 2:
x^2 = 9^2 = 81
y^2 = 2^2 = 4
The difference of these squares is 81 - 4 = 77, which is equal to the given number.
Therefore, the pair of factors for 77 using the difference of two squares is (9+2)(9-2) = 11 x 7.
c. For 112:
Let's try to find two numbers whose squares differ by 112, so we can represent the squares as (x^2) and (y^2).
Starting with x = 11 and y = 5:
x^2 = 11^2 = 121
y^2 = 5^2 = 25
The difference of these squares is 121 - 25 = 96, which is not equal to 112.
Now let's try x = 14 and y = 2:
x^2 = 14^2 = 196
y^2 = 2^2 = 4
The difference of these squares is 196 - 4 = 192, which is not equal to 112.
Since we couldn't find two numbers whose squares differ by 112, we don't have a pair of factors for 112 using the difference of two squares.
To find a pair of factors using the difference of two squares, we need to identify a perfect square number that is less than the given number. Then, we can express the given number as the difference between two perfect squares.
a. For the number 45, let's find a perfect square number less than 45. The largest perfect square less than 45 is 36 (which is 6^2). Now, we can express 45 as the difference of two perfect squares by using the formula: a^2 - b^2 = (a + b)(a - b). In this case, a is 6 and b is unknown. So, we have 45 = (6 + b)(6 - b).
We can further solve this equation by trying different values for b. By substituting different values of b, we find that when b = 3, the equation becomes 45 = (6 + 3)(6 - 3). So, the pair of factors for 45 using the difference of two squares are (6 + 3) and (6 - 3). This gives us the factors 9 and 3, since (6 + 3) = 9 and (6 - 3) = 3.
Therefore, the pair of factors for 45 using the difference of two squares is: 9 and 3.
b. Similarly, for the number 77, let's find a perfect square number less than 77. The largest perfect square less than 77 is 64 (which is 8^2). Now, we can express 77 as the difference of two perfect squares by using the formula: a^2 - b^2 = (a + b)(a - b). In this case, a is 8 and b is unknown. So, we have 77 = (8 + b)(8 - b).
Again, we can solve this equation by trying different values for b. By substituting different values of b, we find that when b = 1, the equation becomes 77 = (8 + 1)(8 - 1). So, the pair of factors for 77 using the difference of two squares are (8 + 1) and (8 - 1). This gives us the factors 9 and 7, since (8 + 1) = 9 and (8 - 1) = 7.
Therefore, the pair of factors for 77 using the difference of two squares is: 9 and 7.
c. For the number 112, let's find a perfect square number less than 112. The largest perfect square less than 112 is 100 (which is 10^2). Now, we can express 112 as the difference of two perfect squares by using the formula: a^2 - b^2 = (a + b)(a - b). In this case, a is 10 and b is unknown. So, we have 112 = (10 + b)(10 - b).
Again, we can solve this equation by trying different values for b. By substituting different values of b, we find that when b = 2, the equation becomes 112 = (10 + 2)(10 - 2). So, the pair of factors for 112 using the difference of two squares are (10 + 2) and (10 - 2). This gives us the factors 12 and 8, since (10 + 2) = 12 and (10 - 2) = 8.
Therefore, the pair of factors for 112 using the difference of two squares is: 12 and 8.