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Solve -3x² - 3x - 1 = 0

-3x² - 3x - 1 = 0 has no real solutions. The two complex solutions are x = (-3 + √3 i)/6 and x = (-3 - √3 i)/6.

x² +x += 0
−3x2−3x−1=0⇒x=−3+3 i6, −3−3 i6 (complex)-3x^2 - 3x - 1 = 0 \Rightarrow x = \frac{-3 + \sqrt{3}\, i}{6},\ \frac{-3 - \sqrt{3}\, i}{6}\ \text{(complex)}
Discriminant D = -3

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

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Graph of y = -3x² - 3x - 1-4-3-2-1123-50-40-30-20-10vertexy = -3x² - 3x - 1
Graph of y = -3x² - 3x - 1. The curve never reaches the x-axis, which is why there are no real solutions; the vertex is at (-0.5, -0.25).

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 = -3, c = -1.

    −3x2−3x−1=0-3x^2 - 3x - 1 = 0
  2. 2

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

    D=(−3)2−4⋅(−3)⋅(−1)=9−12=−3D = (-3)^2 - 4\cdot(-3)\cdot(-1) = 9 - 12 = -3
  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±−3−6x = \frac{-b \pm \sqrt{D}}{2a} = \frac{3 \pm \sqrt{-3}}{-6}
  5. 5

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

    x=−3+3 i6orx=−3−3 i6x = \frac{-3 + \sqrt{3}\, i}{6}\quad\text{or}\quad x = \frac{-3 - \sqrt{3}\, i}{6}
LaTeX source of the formulas on this page
\[ -3x^2 - 3x - 1 = 0\ \Rightarrow\ x = \frac{-3 + \sqrt{3}\, i}{6},\ \frac{-3 - \sqrt{3}\, i}{6} \]
\[ -3x^2 - 3x - 1 = 0 \]
\[ D = (-3)^2 - 4\cdot(-3)\cdot(-1) = 9 - 12 = -3 \]
\[ x = \frac{-b \pm \sqrt{D}}{2a} = \frac{3 \pm \sqrt{-3}}{-6} \]
\[ x = \frac{-3 + \sqrt{3}\, i}{6}\quad\text{or}\quad x = \frac{-3 - \sqrt{3}\, i}{6} \]

The parabola y = -3x² - 3x - 1

xy
-3.5-27.25
-2.5-12.25
-1.5-3.25
-0.5-0.25
0.5-3.25
1.5-12.25
2.5-27.25

Questions

What are the solutions of -3x² - 3x - 1 = 0?

-3x² - 3x - 1 = 0 has no real solutions. The two complex solutions are x = (-3 + √3 i)/6 and x = (-3 - √3 i)/6.

Does -3x² - 3x - 1 = 0 factor?

Not over the integers, because the discriminant -3 is negative. Use the quadratic formula.

What is the vertex of y = -3x² - 3x - 1?

(-1/2, -1/4). 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 = 1/3.

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