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Solve -3x² - 3x - 11 = 0

-3x² - 3x - 11 = 0 has no real solutions. The two complex solutions are x = (-3 + √123 i)/6 and x = (-3 - √123 i)/6.

x² +x += 0
−3x2−3x−11=0⇒x=−3+123 i6, −3−123 i6 (complex)-3x^2 - 3x - 11 = 0 \Rightarrow x = \frac{-3 + \sqrt{123}\, i}{6},\ \frac{-3 - \sqrt{123}\, i}{6}\ \text{(complex)}
Discriminant D = -123

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

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Graph of y = -3x² - 3x - 11-4-3-2-1123-60-50-40-30-20-10vertexy = -3x² - 3x - 11
Graph of y = -3x² - 3x - 11. The curve never reaches the x-axis, which is why there are no real solutions; the vertex is at (-0.5, -10.25).

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 = -11.

    −3x2−3x−11=0-3x^2 - 3x - 11 = 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)⋅(−11)=9−132=−123D = (-3)^2 - 4\cdot(-3)\cdot(-11) = 9 - 132 = -123
  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±−123−6x = \frac{-b \pm \sqrt{D}}{2a} = \frac{3 \pm \sqrt{-123}}{-6}
  5. 5

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

    x=−3+123 i6orx=−3−123 i6x = \frac{-3 + \sqrt{123}\, i}{6}\quad\text{or}\quad x = \frac{-3 - \sqrt{123}\, i}{6}
LaTeX source of the formulas on this page
\[ -3x^2 - 3x - 11 = 0\ \Rightarrow\ x = \frac{-3 + \sqrt{123}\, i}{6},\ \frac{-3 - \sqrt{123}\, i}{6} \]
\[ -3x^2 - 3x - 11 = 0 \]
\[ D = (-3)^2 - 4\cdot(-3)\cdot(-11) = 9 - 132 = -123 \]
\[ x = \frac{-b \pm \sqrt{D}}{2a} = \frac{3 \pm \sqrt{-123}}{-6} \]
\[ x = \frac{-3 + \sqrt{123}\, i}{6}\quad\text{or}\quad x = \frac{-3 - \sqrt{123}\, i}{6} \]

The parabola y = -3x² - 3x - 11

xy
-3.5-37.25
-2.5-22.25
-1.5-13.25
-0.5-10.25
0.5-13.25
1.5-22.25
2.5-37.25

Questions

What are the solutions of -3x² - 3x - 11 = 0?

-3x² - 3x - 11 = 0 has no real solutions. The two complex solutions are x = (-3 + √123 i)/6 and x = (-3 - √123 i)/6.

Does -3x² - 3x - 11 = 0 factor?

Not over the integers, because the discriminant -123 is negative. Use the quadratic formula.

What is the vertex of y = -3x² - 3x - 11?

(-1/2, -41/4). 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 = 11/3.

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