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

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.


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