What 2 numbers add to make 10 and multiply to make 40?

Answer 1

First we have to set up our two equations:

X + Y = 10 XY = 40

We solve the first equation for Y: X + Y = 10 => X + Y - X = 10 - X => Y = 10 - X

Substitute the Y value in the first equation into the second equation:

XY = 40 => X(10 - X) = 40 => 10X - X^2 = 40

Rearrange the terms to produce a quadratic: -X^2 + 10X = 40 => -X^2 + 10X - 40 = 40 - 40 => -X^2 + 10X - 40 = 0

Flip the signs: -X^2 + 10X - 40 = 0 => +X^2 - 10X + 40 = 0 => X^2 - 10X + 40 = 0

Here, coefficient of X^2 is 1, X is -10 and the constant is 40. So, a = 1, b = -10, c = 40.

Substitute these values into the quadratic formula to find X: X = [-b ± √(b^2 - 4ac)]/2a => X = [-(-10) ± √((-10)^2 - 4(1)(40)]/2(1) => X = [10 ± √(100 - 160)]/2 => X = [10 ± √(-60)]/2 => X = [10 ± √(-1 * 2 * 2 * 3 * 5)]/2 => X = [10 ± √(-1) * √2 * √2 * √3 * √5]/2 => X = [10 ± i * 2 * √3 * √5]/2 => X = [10 ± 2√(3*5)i]/2] => X = (10 ± 2√15i)/2 => X = 10/2 ± 2√15i/2 => X = 5 ± √15i X = 5 + √15i or X = 5 - √15i

Substitute these into the first equation to find their respective Y values.

For X = 5 + √15i: Y = 10 - X => Y = 10 - (5 + √15i) => Y = 10 - 5 - √15i => Y = 5 - √15i

For X = 5 - √15i; Y = 10 - X => Y = 10 - (5 - √15i) => Y = 10 - 5 + √15i => Y = 5 - √15i

We see that the X in equation two equals the Y in equation 1 and vice versa. So these are the only two values.

So, the two numbers are 5 - √15i and 5 + √15i.

It was quite a famous question, being featured on the Area Magne as an unsolvable question because of the methods at the time used to solve equations not being equipped for complex, even negative numbers. Including famous methods such as Cardano's Method, Geometric methods, etc. Nowadays it would be quite an easy question involving very basic use of the imaginary number(i = √-1), and a knowledge of how to solve linear systems of equations(a fancy way of saying a bunch of related equations). But it was enough to stump math geniuses of that time. Now that I think about it, a lot of high school math today would stump mathematicians of old. But, we still owe it to them.

Hope this helps :)

Answer 2

It can not be done

The multiples of 40 are...


1 and 40, which add up to 41


2 and 20, which add up to 22


4 and 10, which add up to 14


and


5 and 8, which add up to 13

Answer 3

5 plus square root of minus15 & 5 minus square root of minus 15.

Answer 4

5×8