Question

What two numbers multiply to -126 and add up to -65

Answer 1

Xy = -126
x + y = -65

$xy = -126$
$\frac{xy}{x} = \frac{-126}{x}$
$y = \frac{-126}{x}$

$x + y = -65$
$x + \frac{-126}{x} = -65$
$\frac{x^{2}}{x} + \frac{-126}{x} = -65$
$\frac{x^{2} - 126}{x} = -65$
$-65x = x^{2} - 126$
$0 = x^{2} + 65x - 126$
$x = \frac{-(65) \+ \sqrt{(65)^{2} - 4(1)(-126)}}{2(1)}$
$x = \frac{-65 \± \sqrt{4225 + 504}}{2}$
$x = \frac{-65 \± \sqrt{4729}}{2}$
$x = \frac{-65 \± 68.8}{2}$
$x = \frac{-65 + 68.8}{2}$  $or$  $x = \frac{-65 - 68.8}{2}$
$x = \frac{3.8}{2}$  $or$  $x = \frac{-133.8}{2}$
$x = 1.9$  $or$  $x = -66.9$

     x + y = -65
  1.9 + y = -65
- 1.9           - 1.9
           y = -66.9
    (x, y) = (1.9, -66.9)

or

        x + y = -65
  -66.9 + y = -65
+ 66.9        + 66.9
              y = 1.9
        (x, y) = (-66.9, 1.9)

The two numbers that add up to -65 and can multiply to -126 are the numbers -66.9 and 1.9.