What adds to -12 and multiplies to 34?

Answer 1

The two numbers are: -7.414 & -4.586

I will assume we are looking for 2 numbers only here.

1. x + y = -12

2. x * y = 34

We can rewrite 1. as:

3. y = -x - 12

We can now substitute this value for y back into 2. to give:

4. x * (-x - 12) = 34

We can now expand the brackets:

-x2 - 12x = 34

Now we write this as an equation equal to 0 to put it into quadratic form:

-x2 - 12x - 34 = 0

Now we can use the quadratic formula to find the roots:

Where ax2 + bx + c = 0 then:

x = (-b + or - ((b2 - 4ac)1/2)) / 2a

Substituting in the values in -x2 - 12x - 34 = 0 gives:

x = (--12 + or - ((-122 - (4 * -1 * -34))1/2)) / (2 * -1)

x = (12 + or - (144 - 136)1/2) / -2

x = (12 + or - 81/2) / -2

We can now calculate the above to the desired level of accuracy to solve for the two roots of the quadratic equation. These roots will be the numbers we seek.

I will answer exact only to 3 decimal places (for my own convenience).

x = -7.414 & -4.586

(Note 1: These two numbers will not multiply to give exactly 34 as they are only accurate to 3 decimal places; the more accurately we answer the question the closer to 34 the answers will be when multiplied.)

(Note 2: x1/2 is an alternate way of writing "the square root of x".)