Question

What two numbers multiply to 54 and add up to 7

Answer 1

xy = 54
x + y = 7


`xy = 54`

`(xy)/(x) = (54)/(x)`

`y = (54)/(x)`


`x + y = 7`

`x + (54)/(x) = 7`

`(x^(2))/(x) + (54)/(x) = 7`

`(x^(2) + 54)/(x) = 7`

`x^(2) + 54 = 7x`

`x^(2) - 7x + 54 = 0`

`x = \frac{-(-7) \± \sqrt{(-7)^(2) - 4(1)(54)}}{2(1)}`

`x = (7 \± √(49 - 216))/(2)`

`x = (7 \± √(-167))/(2)`

`x = (7 \± i√(167))/(2)`

`x = (7 \± 12.9i)/(2)`

`x = 3.5 \± 6.45i`

`x = 3.5 + 6.45i`
`or` 
`3.5 - 6.45i`

x + y = 7
3.5 + 6.45i + y = 7
- (3.5 + 6.45i) - (3.5 + 6.45i)
y = 3.5 - 6.45i
(x, y) = (3.5 + 6.45i, (3.5 - 6.45i)

or

x + y = 7
3.5 - 6.45i + y = 7
- (3.5 - 6.45i) - (3.5 - 6.45i)
y = 3.5 + 6.45i
(x, y) = (3.5 - 6.45i, 3.5 + 6.45i)

The two numbers that add up to 7 and can multiply to 54 is 3.5 ± 6.45i.