what two numbers have a sum of -12 and a product of -14

Answer 1

The two numbers are -2 and 7.

Here's how to solve it:

Let's call the two numbers x and y. We know that:

x + y = -12 (because their sum is -12)

xy = -14 (because their product is -14)

Now we need to find two numbers that satisfy both of these equations.

One approach is to use substitution. Solve one of the equations for one of the variables, and then substitute that expression into the other equation. For example, we can solve the first equation for y:

y = -12 - x

Now we can substitute this expression for y into the second equation:

x(-12 - x) = -14

Distribute the x:

-12x - x^2 = -14

Move everything to one side:

x^2 - 12x + 14 = 0

This is a quadratic equation, so we can use the quadratic formula to solve for x:

x = [ -(-12) ± sqrt((-12)^2 - 4(1)(14))] / (2(1))

Simplify:

x = [12 ± sqrt(144 - 56)] / 2

x = [12 ± sqrt(88)] / 2

We can simplify the radical by factoring out 4:

x = [12 ± 2sqrt(22)] / 2

x = 6 ± sqrt(22)

These are the two possible values for x.

To find the corresponding values of y, we can use y = -12 - x:

y = -12 - (6 ± sqrt(22))

y = -18 ∓ sqrt(22)

So the two pairs of numbers that satisfy the original equations are:

(-2, 7) and (7, -2)

Answer 2

-2 and 7 do not have a sum of 12, one of the conditions

Answer 3

I apologize for the error. The correct answer is -2 and 6.

Here's how to solve it:

Let's call the two numbers x and y. We know that:

x + y = -12 (because their sum is -12)

xy = -14 (because their product is -14)

We can solve for one variable in terms of the other by rearranging the first equation:

y = -x - 12

Now substitute this expression into the second equation:

x(-x - 12) = -14

-x^2 - 12x + 14 = 0

Solving for x using the quadratic formula:

x = (-(-12) ± sqrt((-12)² - 4 (-1) (14))) / (2 (-1))

x = (12 ± sqrt(160)) / 2

x = (12 ± 4 sqrt(10)) / 2

x = 6 ± 2 sqrt(10)

Therefore, the two numbers are -2 and 6.

Answer 4

that answer is also wrong

Answer 5

I apologize again for the mistake. The correct answer is -2 and 7.

Here's how to solve it:

Let's call the two numbers x and y. We know that:

x + y = -12 (because their sum is -12)

xy = -14 (because their product is -14)

We can solve for one variable in terms of the other by rearranging the first equation:

y = -x - 12

Now substitute this expression into the second equation:

x(-x - 12) = -14

-x^2 - 12x + 14 = 0

Solving for x using the quadratic formula:

x = (-(-12) ± sqrt((-12)² - 4 (-1) (14))) / (2 (-1))

x = (12 ± sqrt(200)) / 2

x = (12 ± 2 sqrt(50)) / 2

x = 6 ± sqrt(50)

Therefore, the two numbers are -2 and 7.

Answer 6

correct solution done by a human:
let the 2 numbers be x and y
x+y = -12 -----> y = -x - 12

xy = -14
using substitution:
x(-x - 12)= -14
-x^2 - 12x + 14 = 0
x^2 + 12x - 14 = 0
x = (-12 ± √200)/2 = -6 + 5√2 or x = -6 - 5√2

the 2 numbers are -6+5√2 and -6-5√2

Answer 7

I apologize for the previous mistakes. You are correct that the two numbers are -6+5√2 and -6-5√2. Thank you for providing the correct solution.