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Solve -3x² + 5x - 6 = 0

-3x² + 5x - 6 = 0 has no real solutions. The two complex solutions are x = (5 + √47 i)/6 and x = (5 - √47 i)/6.

x² +x += 0
−3x2+5x−6=0⇒x=5+47 i6, 5−47 i6 (complex)-3x^2 + 5x - 6 = 0 \Rightarrow x = \frac{5 + \sqrt{47}\, i}{6},\ \frac{5 - \sqrt{47}\, i}{6}\ \text{(complex)}
Discriminant D = -47

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

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Graph of y = -3x² + 5x - 6-3-2-11234-50-40-30-20-10vertexy = -3x² + 5x - 6
Graph of y = -3x² + 5x - 6. The curve never reaches the x-axis, which is why there are no real solutions; the vertex is at (0.83333333, -3.9166667).

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 = 5, c = -6.

    −3x2+5x−6=0-3x^2 + 5x - 6 = 0
  2. 2

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

    D=52−4⋅(−3)⋅(−6)=25−72=−47D = 5^2 - 4\cdot(-3)\cdot(-6) = 25 - 72 = -47
  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=−5±−47−6x = \frac{-b \pm \sqrt{D}}{2a} = \frac{-5 \pm \sqrt{-47}}{-6}
  5. 5

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

    x=5+47 i6orx=5−47 i6x = \frac{5 + \sqrt{47}\, i}{6}\quad\text{or}\quad x = \frac{5 - \sqrt{47}\, i}{6}
LaTeX source of the formulas on this page
\[ -3x^2 + 5x - 6 = 0\ \Rightarrow\ x = \frac{5 + \sqrt{47}\, i}{6},\ \frac{5 - \sqrt{47}\, i}{6} \]
\[ -3x^2 + 5x - 6 = 0 \]
\[ D = 5^2 - 4\cdot(-3)\cdot(-6) = 25 - 72 = -47 \]
\[ x = \frac{-b \pm \sqrt{D}}{2a} = \frac{-5 \pm \sqrt{-47}}{-6} \]
\[ x = \frac{5 + \sqrt{47}\, i}{6}\quad\text{or}\quad x = \frac{5 - \sqrt{47}\, i}{6} \]

The parabola y = -3x² + 5x - 6

xy
-2.1667-30.917267
-1.1667-15.917067
-0.1667-6.9168667
0.8333-3.9166667
1.8333-6.9164667
2.8333-15.916267
3.8333-30.916067

Questions

What are the solutions of -3x² + 5x - 6 = 0?

-3x² + 5x - 6 = 0 has no real solutions. The two complex solutions are x = (5 + √47 i)/6 and x = (5 - √47 i)/6.

Does -3x² + 5x - 6 = 0 factor?

Not over the integers, because the discriminant -47 is negative. Use the quadratic formula.

What is the vertex of y = -3x² + 5x - 6?

(5/6, -47/12). The parabola opens downwards, so this is the maximum.

What are the sum and product of the roots?

Sum = −b/a = 5/3, product = c/a = 2.

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