Question

What adds up to -54 and multiplies to -16

Answer 1

x + y = -54
xy = -16

$xy = -16$
$\frac{xy}{x} = \frac[-16}{x}$
$y = \frac{-16}{x}$

$x + y = -54$
$x + \frac{-16}{x} = -54$
$\frac{x^{2}}{x} + \frac{-16}{x} = -54$
$\frac{x^{2} - 16}{x} = -54$
$x^{2} - 16 = -54x$
$x^{2} + 54x - 16 = 0$
$x = \frac{-(54) \± \sqrt{(54)^{2} - 4(1)(-16)}}{2(1)}$
$x = \frac{-54 \± \sqrt{2916 + 64}}{2}$
$x = \frac{-54 \± \sqrt{2980}}{2}$
$x = \frac{-54 \± 59.6}{2}$
$x = -27 \± 29.8$
$x = -27 + 29.8$    $or$    $x = -27 - 29.8$
$x = 2.8$    $or$    $x = 56.8$

     x + y = -54
  2.8 + y = -54
- 2.8          - 2.8
           y = -56.8
     (x, y) = (2.8, -56.8)

             or

        x + y = -54
  -56.8 + y = -54
+ 56.8        + 56.8
              y = 2.8
        (x, y) = (-56.8, 2.8)

The two numbers that add up to -54 and multiply to -16 are -56.8 and 2.8.