Question

What two numbers add to 5 and multiply to -3

Answer 1

`x = (5)/(2) - (√(37))/(2)` and
`y = (5)/(2) + (√(37))/(2)`

`x = (5)/(2) + (√(37))/(2)` and
`y = (5)/(2) - (√(37))/(2)`

Explanation:

Let the numbers be x and y.

We now have a system of equations.

x + y = 5

xy = -3

Solve the second equation for x.

x = -3/y

Now substitute x in the first equation with -3/y.

-3/y + y = 5

Multiply both sides by y.

-3 + y^2 = 5y

y^2 - 5y - 3 = 0

Use the quadratic formula to solve for y.

`y = (-b \pm √(b^2 - 4ac))/(2a)`

`y = (-(-5) \pm √((-5)^2 - 4(1)(-3)))/(2(1))`

`y = (5 \pm √(25 + 12))/(2)`

`y = (5 \pm √(37))/(2)`

`y = (5 + √(37))/(2)` or
`y = (5 - √(37))/(2)`

`y = (5)/(2) + (√(37))/(2)` or
`y = (5)/(2) - (√(37))/(2)`

We get 2 solutions for y. Now for each solution for y, we need to find a corresponding solution for x.

Solve the first equation for x.

x + y = 5

x = 5 - y

Substitute each y value to find the corresponding x value.

`x = 5 - ((5)/(2) + (√(37))/(2))`

`x = 5 - (5)/(2) - (√(37))/(2)`

`x = (10)/(2) - (5)/(2) - (√(37))/(2)`

`x = (5)/(2) - (√(37))/(2)`

This give one solution as:

`x = (5)/(2) - (√(37))/(2)` and
`y = (5)/(2) + (√(37))/(2)`

Now we substitute the other y value to find the other x value.

`x = 5 - ((5)/(2) - (√(37))/(2))`

`x = 5 - (5)/(2) + (√(37))/(2)`

`x = (10)/(2) - (5)/(2) + (√(37))/(2)`

`x = (5)/(2) + (√(37))/(2)`

This give the second solution as:

`x = (5)/(2) + (√(37))/(2)` and
`y = (5)/(2) - (√(37))/(2)`