What 2 number add up to -19 and multiply by 24?

Answer 1

Let x and y be the unknowns. Based on the given problem, we have:

x + y = -19 [The sum of two numbers is -19]

xy = 24 [The product of two numbers is 24]

Solve for these numbers either by elimination or substitution method. The recommended choice is substitution method.

Solve for y for the second equation:

y = 24/x

Substitute that expression for the first equation:

x + (24/x) = -19

Solve for x by rearranging the terms:

x² + 24 = -19x [Multiply both sides by x]

x² + 19x + 24 = 0

Since the expression can't be factored, you don't get integers for x and y. Let's complete the square, and see what we have:

x² + 19x + (19/2)² = -24 + (19/2)²

(x + (19/2))² = 66.25

x + (19/2) = ±√(66.25) or ±√(265/4)

x + (19/2) = ±√(265)/2

x = (-19 ± √(265))/2

Finally, substitute that value back for either of the equation and solve for y. You should get:

y = ±24/((-19 ± √(265))/2)

= ½(-19 ∓ √265)

So the numbers are:

x = ½ * (-19 - √265) and y = ½ * (√265 - 19)

y = ½ * (√(265) - 19) and y = ½ * (-19 - √265)