What two number multiply to 96 but add up to -27?

Answer 1

69

Answer 2

There are no integers that meet this condition, but that doesn't mean that the numbers we're looking for don't exist. What we'll need to do is assemble what we're told into a quadratic equation, and solve from there.

Let's call our numbers x and y. We're told:

x + y = -27

xy = 96

Let's do a quick substitution here and solve for one variable:

x + (96/x) = -27

x2 + 96 = -27x

x2 + 27x = -96

x2 + 27x + (27/2)2 = (27/2)2 - 96

(x + 27/2)2 = 729/4 - 384/4

x + 27/2 = ± √(345 / 4)

x + 27/2 = ± √345 / 2

x = (-27 ± √345) / 2

Now we can simply do the originally stated operations with our two values of x to check to see if our answer is correct. Let's multiply them first:

[ (-27 + √345) / 2 ] * [ (-27 - √345) / 2 ]

= [(-27)2 - 345] / 4

= (729 - 345) / 4

= 384 / 4

= 96

And now we'll add them:

(-27 + √345) / 2 + (-27 - √345) / 2

= (-27 + √345 - 27 - √345) / 2

= (-27 - 27) / 2

= -27

Which confirms that our answer is correct.