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The difference between two numbers is 10 and their product is 3000. What are the two numbers?

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David Shabazi answered

The difference between two numbers is 10 and their product is 3000. The difference between two numbers is 10

Which would mean x - y = 10, because x and y are two different numbers and "difference" is a keyword for subtracting.
Their product is 3000

Which would mean x * y = 3000, because "product" is a keyword for "multiplying".

Now we have two equations we can use to obtain the values of x and y, which are: X - y = 10 and x * y = 3000.

Let's take the first equation and solve for x, and then solve for y.

x - y = 10

x = 10 + y

y = x - 10

Now let's plug these in into the multiplication equation. But we need to only plug one variable at a time. Observe:

x * y = 3000

x * (x - 10) = 3000

x^2 - 10x = 3000

x^2 - 10x - 3000 = 0

To solve for x here, we need to use the quadratic formula, which is x=frac{-bpmsqrt{b^2-4ac }}{2a}. B represents the constant in front of the "x", A stands for the constant in front of the "x^2", and C stands for just the constant without any variable (which would be the -3000).

x = {10 ± √(-10)^2 - 4(1)(-3000)}/2(1)

x = (10 ± √100 + 12000)/2

x = (10 ± √12100)/2

x = (10 ± 110)/2

We have two values for x:

x = (10 + 110)/2 = 120/2 = 60

x = (10 - 110)/2 = -100/2 = -50

Now that we know what x could be, let's plug them back into the original equations.

x = 10 + y

60 = 10 + y

50 = y (Therefore, if x is 60, y must be 50. And it works. Plug these numbers into the equations and they will be equal.)

However, we had a second number for x. Let's plug that in.

x = 10 + y

-50 = 10 + y

-60 = y (If x = -50, then y must be -60. These also work for the original equations when you plug them in, mainly because multiplying two negatives together cancel out the negative signs.)

So you can have: x = 60 and y = 50 OR x = -50 and y = -60.

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