Question

The product of two numbers is 1856. The sum is ninety. What is the difference of these two numbers?

Answer 1

Answer:26

Step-by-step explanation:

x+y=90

xy=1856

x*(90-x)=1856

-x^2 +90x -1856=0

or x^2 -90x +1856=0

Using quadratic equation

(x-58)*(x-32)=0

x=58, y=32

58-32=26

Answer 2

Answer: 26

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Step-by-step explanation:

x and y are the two numbers

The product of those values is 1856, so xy = 1856

The sum is 90, so x+y = 90. Solve for y to get y = 90-x

Plug this into the first equation

xy = 1856

x(90-x) = 1856

90x-x^2 = 1856

0 = x^2-90x+1856

x^2-90x+1856 = 0

Now use the quadratic formula

`x = (-b\pm√(b^2-4ac))/(2a)\\\\x = (-(-90)\pm√((-90)^2-4(1)(1856)))/(2(1))\\\\x = (90\pm√(676))/(2)\\\\x = (90\pm26)/(2)\\\\x = (90+26)/(2) \ \text{ or } \ x = (90-26)/(2)\\\\x = (116)/(2) \ \text{ or } \ x = (64)/(2)\\\\x = 58 \ \text{ or } \ x = 32\\\\`

If x = 58, then,

y = 90-x

y = 90-58

y = 32

So one pair of values is (x,y) = (58,32)

We see that x*y = 58*32 = 1856

If x = 32, then,

y = 90-x

y = 90-32

y = 58

Showing (x,y) = (32,58) is the other solution. But this is just a mirror copy of the first solution. Effectively we're saying the same thing.

The two numbers 58 and 32 multiply to 1856, and they add to 90.

The difference in the two values is 58-32 = 26 which is the final answer.

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In summary,

58*32 = 1856

58+32 = 90

58 - 32 = 26