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Solve -3x² - 2x - 8 = 0

-3x² - 2x - 8 = 0 has no real solutions. The two complex solutions are x = (-1 + √23 i)/3 and x = (-1 - √23 i)/3.

x² +x += 0
−3x2−2x−8=0⇒x=−1+23 i3, −1−23 i3 (complex)-3x^2 - 2x - 8 = 0 \Rightarrow x = \frac{-1 + \sqrt{23}\, i}{3},\ \frac{-1 - \sqrt{23}\, i}{3}\ \text{(complex)}
Discriminant D = -92

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

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Graph of y = -3x² - 2x - 8-4-3-2-1123-50-40-30-20-10vertexy = -3x² - 2x - 8
Graph of y = -3x² - 2x - 8. The curve never reaches the x-axis, which is why there are no real solutions; the vertex is at (-0.33333333, -7.6666667).

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 = -8.

    −3x2−2x−8=0-3x^2 - 2x - 8 = 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)⋅(−8)=4−96=−92D = (-2)^2 - 4\cdot(-3)\cdot(-8) = 4 - 96 = -92
  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±−92−6x = \frac{-b \pm \sqrt{D}}{2a} = \frac{2 \pm \sqrt{-92}}{-6}
  5. 5

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

    x=−1+23 i3orx=−1−23 i3x = \frac{-1 + \sqrt{23}\, i}{3}\quad\text{or}\quad x = \frac{-1 - \sqrt{23}\, i}{3}
LaTeX source of the formulas on this page
\[ -3x^2 - 2x - 8 = 0\ \Rightarrow\ x = \frac{-1 + \sqrt{23}\, i}{3},\ \frac{-1 - \sqrt{23}\, i}{3} \]
\[ -3x^2 - 2x - 8 = 0 \]
\[ D = (-2)^2 - 4\cdot(-3)\cdot(-8) = 4 - 96 = -92 \]
\[ x = \frac{-b \pm \sqrt{D}}{2a} = \frac{2 \pm \sqrt{-92}}{-6} \]
\[ x = \frac{-1 + \sqrt{23}\, i}{3}\quad\text{or}\quad x = \frac{-1 - \sqrt{23}\, i}{3} \]

The parabola y = -3x² - 2x - 8

xy
-3.3333-34.666067
-2.3333-19.666267
-1.3333-10.666467
-0.3333-7.6666667
0.6667-10.666867
1.6667-19.667067
2.6667-34.667267

Questions

What are the solutions of -3x² - 2x - 8 = 0?

-3x² - 2x - 8 = 0 has no real solutions. The two complex solutions are x = (-1 + √23 i)/3 and x = (-1 - √23 i)/3.

Does -3x² - 2x - 8 = 0 factor?

Not over the integers, because the discriminant -92 is negative. Use the quadratic formula.

What is the vertex of y = -3x² - 2x - 8?

(-1/3, -23/3). 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 = 8/3.

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