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Solve -3x² - 2x - 9 = 0

-3x² - 2x - 9 = 0 has no real solutions. The two complex solutions are x = (-1 + √26 i)/3 and x = (-1 - √26 i)/3.

x² +x += 0
−3x2−2x−9=0⇒x=−1+26 i3, −1−26 i3 (complex)-3x^2 - 2x - 9 = 0 \Rightarrow x = \frac{-1 + \sqrt{26}\, i}{3},\ \frac{-1 - \sqrt{26}\, i}{3}\ \text{(complex)}
Discriminant D = -104

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

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Graph of y = -3x² - 2x - 9-4-3-2-1123-60-50-40-30-20-10vertexy = -3x² - 2x - 9
Graph of y = -3x² - 2x - 9. The curve never reaches the x-axis, which is why there are no real solutions; the vertex is at (-0.33333333, -8.6666667).

How to work it out

  1. 1

    Write it in standard form a x² + b x + c = 0 and read off the coefficients: a = -3, b = -2, c = -9.

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

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

    D=(−2)2−4⋅(−3)⋅(−9)=4−108=−104D = (-2)^2 - 4\cdot(-3)\cdot(-9) = 4 - 108 = -104
  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=2±−104−6x = \frac{-b \pm \sqrt{D}}{2a} = \frac{2 \pm \sqrt{-104}}{-6}
  5. 5

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

    x=−1+26 i3orx=−1−26 i3x = \frac{-1 + \sqrt{26}\, i}{3}\quad\text{or}\quad x = \frac{-1 - \sqrt{26}\, i}{3}
LaTeX source of the formulas on this page
\[ -3x^2 - 2x - 9 = 0\ \Rightarrow\ x = \frac{-1 + \sqrt{26}\, i}{3},\ \frac{-1 - \sqrt{26}\, i}{3} \]
\[ -3x^2 - 2x - 9 = 0 \]
\[ D = (-2)^2 - 4\cdot(-3)\cdot(-9) = 4 - 108 = -104 \]
\[ x = \frac{-b \pm \sqrt{D}}{2a} = \frac{2 \pm \sqrt{-104}}{-6} \]
\[ x = \frac{-1 + \sqrt{26}\, i}{3}\quad\text{or}\quad x = \frac{-1 - \sqrt{26}\, i}{3} \]

The parabola y = -3x² - 2x - 9

xy
-3.3333-35.666067
-2.3333-20.666267
-1.3333-11.666467
-0.3333-8.6666667
0.6667-11.666867
1.6667-20.667067
2.6667-35.667267

Questions

What are the solutions of -3x² - 2x - 9 = 0?

-3x² - 2x - 9 = 0 has no real solutions. The two complex solutions are x = (-1 + √26 i)/3 and x = (-1 - √26 i)/3.

Does -3x² - 2x - 9 = 0 factor?

Not over the integers, because the discriminant -104 is negative. Use the quadratic formula.

What is the vertex of y = -3x² - 2x - 9?

(-1/3, -26/3). The parabola opens downwards, so this is the maximum.

What are the sum and product of the roots?

Sum = −b/a = -2/3, product = c/a = 3.

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