What two numbers have a produt of 18 anda sumof 9?

Answer 1

Call the two numbers x and y such that x is greater than or equal to y. We are given the following information:

xy=18

x+y=9

Since x+y=9, (x+y)^2=81.

Using the distributive property of multiplication over addition twice,

(x+y)2=(x+y)(x+y)=(x+y)x+(x+y)y=x2+xy+xy+y2=x2+2xy+y2.

Now we have that x2+2xy+y2=81.

Since xy=18,4xy=72. Subtracting 4xy from the left hand side and subtracting 72 from the right hand side gives x2-2xy+y2=9 (This should still be true since I subtracted the same thing from both sides.). However, (x-y)2=x2-2xy+y2, so (x-y)2=9. This gives that x-y=either 3 or -3. Since x is greater than or equal to y, x-y is positive, so x-y must be 3. Adding x+y to the left hand side of the equation and 9 to the right hand side of the equation gives

x-y+x+y=9+3 (This should still be true since I added the same thing to both sides.). Simplifying gives 2x=12.

Dividing both sides by 2 gives x=12/2=6.

Now that we have that one of the numbers is 6, the other number must be 18/6, or 3. Since 6+3=9, the numbers 6 and 3 satisfy both conditions.