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Solve -x² - x - 12 = 0

-x² - x - 12 = 0 has no real solutions. The two complex solutions are x = (-1 + √47 i)/2 and x = (-1 - √47 i)/2.

x² +x += 0
−x2−x−12=0⇒x=−1+47 i2, −1−47 i2 (complex)-x^2 - x - 12 = 0 \Rightarrow x = \frac{-1 + \sqrt{47}\, i}{2},\ \frac{-1 - \sqrt{47}\, i}{2}\ \text{(complex)}
Discriminant D = -47

Change a, b or c (whole numbers, a not 0) and the exact solutions follow.

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Graph of y = -x² - x - 12-4-3-2-1123-28-26-24-22-20-18-16-14-12-10vertexy = -x² - x - 12
Graph of y = -x² - x - 12. The curve never reaches the x-axis, which is why there are no real solutions; the vertex is at (-0.5, -11.75).

How to work it out

  1. 1

    Write it in standard form a x² + b x + c = 0 and read off the coefficients: a = -1, b = -1, c = -12.

    −x2−x−12=0-x^2 - x - 12 = 0
  2. 2

    Work out the discriminant D = b² − 4ac. Its sign tells you how many real solutions there are.

    D=(−1)2−4⋅(−1)⋅(−12)=1−48=−47D = (-1)^2 - 4\cdot(-1)\cdot(-12) = 1 - 48 = -47
  3. 3

    D is negative, so there are no real solutions; the two solutions are complex numbers.

  4. 4

    The quadratic formula works for every quadratic:

    x=−b±D2a=1±−47−2x = \frac{-b \pm \sqrt{D}}{2a} = \frac{1 \pm \sqrt{-47}}{-2}
  5. 5

    Simplify (pull the square out of the root) to get the solutions:

    x=−1+47 i2orx=−1−47 i2x = \frac{-1 + \sqrt{47}\, i}{2}\quad\text{or}\quad x = \frac{-1 - \sqrt{47}\, i}{2}
LaTeX source of the formulas on this page
\[ -x^2 - x - 12 = 0\ \Rightarrow\ x = \frac{-1 + \sqrt{47}\, i}{2},\ \frac{-1 - \sqrt{47}\, i}{2} \]
\[ -x^2 - x - 12 = 0 \]
\[ D = (-1)^2 - 4\cdot(-1)\cdot(-12) = 1 - 48 = -47 \]
\[ x = \frac{-b \pm \sqrt{D}}{2a} = \frac{1 \pm \sqrt{-47}}{-2} \]
\[ x = \frac{-1 + \sqrt{47}\, i}{2}\quad\text{or}\quad x = \frac{-1 - \sqrt{47}\, i}{2} \]

The parabola y = -x² - x - 12

xy
-3.5-20.75
-2.5-15.75
-1.5-12.75
-0.5-11.75
0.5-12.75
1.5-15.75
2.5-20.75

Questions

What are the solutions of -x² - x - 12 = 0?

-x² - x - 12 = 0 has no real solutions. The two complex solutions are x = (-1 + √47 i)/2 and x = (-1 - √47 i)/2.

Does -x² - x - 12 = 0 factor?

Not over the integers, because the discriminant -47 is negative. Use the quadratic formula.

What is the vertex of y = -x² - x - 12?

(-1/2, -47/4). The parabola opens downwards, so this is the maximum.

What are the sum and product of the roots?

Sum = −b/a = -1, product = c/a = 12.

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