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Solve -x² - 3x - 9 = 0

-x² - 3x - 9 = 0 has no real solutions. The two complex solutions are x = (-3 + 3√3 i)/2 and x = (-3 - 3√3 i)/2.

x² +x += 0
−x2−3x−9=0⇒x=−3+33 i2, −3−33 i2 (complex)-x^2 - 3x - 9 = 0 \Rightarrow x = \frac{-3 + 3\sqrt{3}\, i}{2},\ \frac{-3 - 3\sqrt{3}\, i}{2}\ \text{(complex)}
Discriminant D = -27

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

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Graph of y = -x² - 3x - 9-5-4-3-2-112-22-20-18-16-14-12-10-8-6vertexy = -x² - 3x - 9
Graph of y = -x² - 3x - 9. The curve never reaches the x-axis, which is why there are no real solutions; the vertex is at (-1.5, -6.75).

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 = -3, c = -9.

    −x2−3x−9=0-x^2 - 3x - 9 = 0
  2. 2

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

    D=(−3)2−4⋅(−1)⋅(−9)=9−36=−27D = (-3)^2 - 4\cdot(-1)\cdot(-9) = 9 - 36 = -27
  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=3±−27−2x = \frac{-b \pm \sqrt{D}}{2a} = \frac{3 \pm \sqrt{-27}}{-2}
  5. 5

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

    x=−3+33 i2orx=−3−33 i2x = \frac{-3 + 3\sqrt{3}\, i}{2}\quad\text{or}\quad x = \frac{-3 - 3\sqrt{3}\, i}{2}
LaTeX source of the formulas on this page
\[ -x^2 - 3x - 9 = 0\ \Rightarrow\ x = \frac{-3 + 3\sqrt{3}\, i}{2},\ \frac{-3 - 3\sqrt{3}\, i}{2} \]
\[ -x^2 - 3x - 9 = 0 \]
\[ D = (-3)^2 - 4\cdot(-1)\cdot(-9) = 9 - 36 = -27 \]
\[ x = \frac{-b \pm \sqrt{D}}{2a} = \frac{3 \pm \sqrt{-27}}{-2} \]
\[ x = \frac{-3 + 3\sqrt{3}\, i}{2}\quad\text{or}\quad x = \frac{-3 - 3\sqrt{3}\, i}{2} \]

The parabola y = -x² - 3x - 9

xy
-4.5-15.75
-3.5-10.75
-2.5-7.75
-1.5-6.75
-0.5-7.75
0.5-10.75
1.5-15.75

Questions

What are the solutions of -x² - 3x - 9 = 0?

-x² - 3x - 9 = 0 has no real solutions. The two complex solutions are x = (-3 + 3√3 i)/2 and x = (-3 - 3√3 i)/2.

Does -x² - 3x - 9 = 0 factor?

Not over the integers, because the discriminant -27 is negative. Use the quadratic formula.

What is the vertex of y = -x² - 3x - 9?

(-3/2, -27/4). The parabola opens downwards, so this is the maximum.

What are the sum and product of the roots?

Sum = −b/a = -3, product = c/a = 9.

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