Take the numbers 4 and 7.
If we square them we get 16 and 49, which added together becomes 65.
If we add 4 and 7 together first and then square we get 121.
The difference is 56. PROVE that there is a connection between the product of 4 and 7 and 56 FOR ANY PAIR OF NUMBERS YOU SELECT.
Take the numbers a and b.
If we square them we get a² and b², which added together becomes a²+b².
If we add a and b together first and then square we get (a+b)².
The difference between (a+b)² and a²+b² is....
Expand and subtract.
is it
a²+b²+2ab-a²+b²
2b²+2ab
if not please help
thanks
or is it
a²+b²+2ab-a²-b²
2ab
That's correct! (the last version).
thanks so much
You're welcome!
To prove the connection between the product of any pair of numbers and 56, we can observe the following pattern:
Let's take any two numbers, x and y.
Step 1: Calculate the product of the two given numbers.
Product = x * y
Step 2: Calculate the sum of the two given numbers.
Sum = x + y
Step 3: Square the sum obtained in Step 2.
Square of Sum = Sum^2 = (x + y)^2
Step 4: Calculate the difference between the square of the sum and the product.
Difference = (x + y)^2 - (x * y)
Now, let's simplify the expression for the difference:
(x + y)^2 - (x * y) = (x^2 + 2xy + y^2) - (x * y) = x^2 + 2xy + y^2 - x * y
We can further simplify the expression as:
(x^2 + y^2) + (2xy - xy) = x^2 + y^2 + xy
By substituting values for x and y, we can see how this expression is related to 56, and hence establish a connection.
For example, let's substitute x = 4 and y = 7 into the expression:
Difference = (4^2 + 7^2) + (4 * 7) = (16 + 49) + 28 = 65 + 28 = 93
We can see that the difference here is not 56, but that's okay. Let's try another example.
Let's substitute x = 2 and y = 9 into the expression:
Difference = (2^2 + 9^2) + (2 * 9) = (4 + 81) + 18 = 85 + 18 = 103
Again, we get a different difference. It seems like the connection with 56 is not valid for any pair of values of x and y.
Therefore, based on the examples and the fact that the difference (x^2 + y^2 + xy) will vary for different pairs of numbers, we cannot prove that there is a connection between the product of any pair of numbers and 56.