The Sum of two numbers is 8.5, and their product is 18. Find the numbers?

Answer 1

Let the 2 numbers be x and y.

Then according to the given question, x + y = 8.5
and xy = 18

First equation gives, y = 8.5 - x --------- (1). Substitute this in the second equation. Which implies x(8.5 - x) = 18.
gives, 8.5x - x^2 = 18
gives, (17/2)x - x^2 = 18 (Taking LCM)
gives, 17x - 2 x^2 = 36
multiplying with "-" on both sides.
gives, 2 x^2 - 17x = -36
gives, 2 x^2 - 17x + 36 = 0
gives, 2 x^2 - 8x - 9x + 36 = 0
Solving this gives x = 4 , 4.5.

Substituting these values in equation 1:

When x = 4. y = 8.5 - 4 = 4.5

When x = 4.5. y = 8.5 - 4.5 = 4

Therefore the two numbers are 4, 4.5

Hope you understood the explanation.

Answer 2

It would be real nice if the numbers were rational numbers... if not, use quadratic equation to solve

x^2 - 8.5 x + 18 = 0

Because the sum is 17/2, we look a half-integer that divide 18.

18 = 1/2 * 36, but sum is 73/2
18 = 1 1/2 * 12, but sum is 27/2
18 = 4 1/2 * 4, and sum is 17/2 BINGO

Answer 3

This is a problem in Freshman Algebra. Start with x+y=8.5 and xy=18. Expand it from there and you should get an answer. It's really a quite simple problem if you remember that whole numbers may not be the answers.

Answer 4

Sherman's answer is truly impressive, but one can determine the answer almost intuitively. I tried algebra, but got stuck. I realized that one of the numbers was fractional and one of the numbers that 18 was divisible by could not produce another integer. That excluded 1, 2 and 3. 4 was the first number I could divide into 18 to reach a nice fraction, which multiplied by another integer, 4, gave the answer.

This was fun.

Answer 5

let the numbers be x and y
x+y = 8.5 and xy = 18

or x + 18/x =8.5
or x^2 - 8.5x + 18 = 0
or x^2 - 4.5x - 4x + 18 = 0
or x(x - 4.5) - 4(x - 4.5) = 0
or (x-4.5)(x-4) = 0
or x = 4.5 and x = 4 or the numbers are 4.5 and 4

Answer 6

x+y=8 and xy=18
x=8-y :::subtract both sides of equation by -y
(8-y)y=18::: substitute x from above
8y-y^2=18 :::distributive property
y^2-8y= - 18 :: multiply both sides by -1
y^2-8y+16= - 18+16 :::complete the square
(y-4)^2 = -2
y-4=+_-2^.5

no real number solution

Answer 7

x + y = 8.5
xy = 18

y = 8.5 - x
x(8.5 - x) = 18
8.5x - x^2 = 18
x^2 - (17/2)x + 18 = 0

x = (-b ± sqrt(b^2 - 4ac))/2a

x = (-(-17/2) ± sqrt((-17/2)^2 - 4(1)(18)))/(2(1))
x = ((17/2) ± sqrt((289/4) - 72))/2
x = ((17/2) ± sqrt((289/4) - (288/4))/2
x = ((17/2) ± sqrt((289 - 288)/4))/2
x = ((17/2) ± sqrt((1/4)))/2
x = ((17/2) ± (1/2))/2
x = ((17/2)/2) ± ((1/2)/2)
x = ((17/2)/(2/1)) ± ((1/2)/(2/1))
x = ((17/2)*(1/2)) ± ((1/2)*(1/2))
x = (17/4) ± (1/4)
x = (17/4) + (1/4) or (17/4) - (1/4)
x = ((17 + 1)/4) or ((17 - 1)/4)
x = (18/4) or (16/4)
x = (9/2) or 4

ANS : 4.5 and 4

Answer 8

2 equations 2 unknowns. Simple algebra. Sounds like a simple homework problem.

Answer 9

x+y=8.5
xy=18; x=18/y

x+y=8.5
18/y+y=8.5
y(18/y+y=8.5)
18+y^2=17y/2
2(18+y^2=17y/2)
36+2y^2=17y
2y^2-17y+36=0
(2y-9)(y-4)=0
2y-9=0 or y-4=0
2y=9 y=4
y=9/2

y=9/2 or y=4

Answer 10

4.5+4=8.5
4.5*4=18