Question

What two numbers multiply to get -36 and add to get 9

Answer 1

xy = -36
x + y = 9

$xy = -36$
$\frac{xy}{x} =\frac{-36}{x}$
$y = \frac{-36}{x}$

$x + y = 9$
$x + \frac{-36}{x} = 9$
$\frac{x^{2}}{x} + \frac{-36}{x} = 9$
$\frac{x^{2} - 36}{x} = 9$
$9x = x^{2} - 36$
$0 = x^{2} - 9x - 36$
$x = \frac{-(-9) \± \sqrt{(-9)^{2} - 4(1)(-36)}}{2(1)}$
$x = \frac{9 \± \sqrt{81 + 144}}{2}$
$x = \frac{9 \± \sqrt{225}}{2}$
$x = \frac{9 \± 15}{2}$
$x = \frac{9 + 15}{2}$    $or$    $x = \frac{9 - 15}{2}$
$x = \frac{24}{2}$    $or$    $x = \frac{-6}{2}$
$x = 12$    $or$    $x = -3$

    x + y = 9       or        x + y = 9
  12 + y = 9      or       -3 + y = 9
- 12      - 12              + 3       + 3
          y = -3         or          y = 12
(x, y) = (12, -3)    or    (x, y) = (-3, 12)

The two numbers that add up to 9 and multiply to -36 are the numbers -3 and 12.