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Solve -3x² + 4x - 11 = 0

-3x² + 4x - 11 = 0 has no real solutions. The two complex solutions are x = (2 + √29 i)/3 and x = (2 - √29 i)/3.

x² +x += 0
−3x2+4x−11=0⇒x=2+29 i3, 2−29 i3 (complex)-3x^2 + 4x - 11 = 0 \Rightarrow x = \frac{2 + \sqrt{29}\, i}{3},\ \frac{2 - \sqrt{29}\, i}{3}\ \text{(complex)}
Discriminant D = -116

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

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Graph of y = -3x² + 4x - 11-3-2-11234-60-50-40-30-20-10vertexy = -3x² + 4x - 11
Graph of y = -3x² + 4x - 11. The curve never reaches the x-axis, which is why there are no real solutions; the vertex is at (0.66666667, -9.6666667).

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 = 4, c = -11.

    −3x2+4x−11=0-3x^2 + 4x - 11 = 0
  2. 2

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

    D=42−4⋅(−3)⋅(−11)=16−132=−116D = 4^2 - 4\cdot(-3)\cdot(-11) = 16 - 132 = -116
  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=−4±−116−6x = \frac{-b \pm \sqrt{D}}{2a} = \frac{-4 \pm \sqrt{-116}}{-6}
  5. 5

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

    x=2+29 i3orx=2−29 i3x = \frac{2 + \sqrt{29}\, i}{3}\quad\text{or}\quad x = \frac{2 - \sqrt{29}\, i}{3}
LaTeX source of the formulas on this page
\[ -3x^2 + 4x - 11 = 0\ \Rightarrow\ x = \frac{2 + \sqrt{29}\, i}{3},\ \frac{2 - \sqrt{29}\, i}{3} \]
\[ -3x^2 + 4x - 11 = 0 \]
\[ D = 4^2 - 4\cdot(-3)\cdot(-11) = 16 - 132 = -116 \]
\[ x = \frac{-b \pm \sqrt{D}}{2a} = \frac{-4 \pm \sqrt{-116}}{-6} \]
\[ x = \frac{2 + \sqrt{29}\, i}{3}\quad\text{or}\quad x = \frac{2 - \sqrt{29}\, i}{3} \]

The parabola y = -3x² + 4x - 11

xy
-2.3333-36.666067
-1.3333-21.666267
-0.3333-12.666467
0.6667-9.6666667
1.6667-12.666867
2.6667-21.667067
3.6667-36.667267

Questions

What are the solutions of -3x² + 4x - 11 = 0?

-3x² + 4x - 11 = 0 has no real solutions. The two complex solutions are x = (2 + √29 i)/3 and x = (2 - √29 i)/3.

Does -3x² + 4x - 11 = 0 factor?

Not over the integers, because the discriminant -116 is negative. Use the quadratic formula.

What is the vertex of y = -3x² + 4x - 11?

(2/3, -29/3). The parabola opens downwards, so this is the maximum.

What are the sum and product of the roots?

Sum = −b/a = 4/3, product = c/a = 11/3.

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