Math help please

Question

What two numbers when added equal 1 and when multiplied equal 12

Answer 1

Let the two numbers be "x" and "y". so:

x + y = 1 and

x•y = 12

Taking the first equation (x + y = 1) and solving it for x, we get y = 1 - x

Plugging "1 - x" into equation two wherever we see a "y", we get:

x•(1 - x) = 12

x - x² = 12 [now, adding x² and subtracting x from both sides of the equation we get:

0 = x² - x + 12

This equation is not factorable so we would need to use the quadratic formula to solve it.

The quadratic formula is:

So a = 1, b = -1, and c = 12 meaning:

x = [-(-1) ± √((-1)² -4(1)(12))]/(2(1)]

x = (1 ± √-47)/2 = 1/2 ± √47/2i

So, in other words there are no REAL numbers that add to 1 and multiply to 12. But there are two complex (imaginary) numbers that do. They are 1/2 + √47/2i and 1/2 - √47/2i

Answer 2

Are there any negatives in this equation? Because otherwise, the number would be an imaginary number. For example, we could set up a system of equations to solve this where:

x+y=1

x*y=12

Then, we can solve for a variable. I will choose y.

y=1-x

Then, we can plug this back into the second equation.

x*(1-x)=12

x-x²=12

We can rearrange this into a quadratic equation

x²-x+12=0

Now, we can use part of the quadratic formula to see whether there are any real roots. If [b²-4ac] is greater than 0, there will be 2 real roots. If it is equal to 0, there is only 1 real root. If it is less that zero, there are no real roots. Therefore

(-1)²-4(1)(12)=-47

-47 is less than zero, so there are no real roots.