Question

which numbers multiply to 50 and add to 6

Answer 1

xy = 50
x + y = 6

$xy = 50$
$\frac{xy}{x} = \frac{50}{x}$
$y = \frac{50}{x}$

$x + y = 6$
$x + \frac{50}{x} = 6$
$\frac{x^{2}}{x} + \frac{50}{x} = 6$
$\frac{x^{2} + 50}{x} = 6$
$x^{2} + 50 = 6x$
$x^{2} - 6x + 50 = 0$
$x = \frac{-(-6) \± \sqrt{(-6)^{2} - 4(1)(50)}}{2(1)}$
$x = \frac{6 \± \sqrt{36 - 100}}{2}$
$x = \frac{6 \± \sqrt{-64}}{2}$
$x = \frac{6 \± 8i}{2}$
$x = 3 \± 4i$
$x = 3 + 4i$    $or$    $x = 3 - 4i$

       x + y = 6
    3 + 4i + y = 6
- (3 + 4i)      - (3 + 4i)
                 y = 3 - 4i
           (x, y) = (3 + 4i, 3 - 4i)

                   or
          x + y = 6
    3 - 4i + y = 6
- (3 - 4i)      - (3 - 4i)
                y = 3 + 4i
          (x, y) = (3 - 4i, 3 + 4i)

(x, y) = (3 ± 4i, 3 ± 4i)

The two numbers that multiply to 50 and add up to 6 are 3 ± 4i.