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Solve -3x² + 9x - 9 = 0

-3x² + 9x - 9 = 0 has no real solutions. The two complex solutions are x = (3 + √3 i)/2 and x = (3 - √3 i)/2.

x² +x += 0
−3x2+9x−9=0⇒x=3+3 i2, 3−3 i2 (complex)-3x^2 + 9x - 9 = 0 \Rightarrow x = \frac{3 + \sqrt{3}\, i}{2},\ \frac{3 - \sqrt{3}\, i}{2}\ \text{(complex)}
Discriminant D = -27

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

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Graph of y = -3x² + 9x - 9-2-112345-50-40-30-20-10vertexy = -3x² + 9x - 9
Graph of y = -3x² + 9x - 9. The curve never reaches the x-axis, which is why there are no real solutions; the vertex is at (1.5, -2.25).

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 = 9, c = -9.

    −3x2+9x−9=0-3x^2 + 9x - 9 = 0
  2. 2

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

    D=92−4⋅(−3)⋅(−9)=81−108=−27D = 9^2 - 4\cdot(-3)\cdot(-9) = 81 - 108 = -27
  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=−9±−27−6x = \frac{-b \pm \sqrt{D}}{2a} = \frac{-9 \pm \sqrt{-27}}{-6}
  5. 5

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

    x=3+3 i2orx=3−3 i2x = \frac{3 + \sqrt{3}\, i}{2}\quad\text{or}\quad x = \frac{3 - \sqrt{3}\, i}{2}
LaTeX source of the formulas on this page
\[ -3x^2 + 9x - 9 = 0\ \Rightarrow\ x = \frac{3 + \sqrt{3}\, i}{2},\ \frac{3 - \sqrt{3}\, i}{2} \]
\[ -3x^2 + 9x - 9 = 0 \]
\[ D = 9^2 - 4\cdot(-3)\cdot(-9) = 81 - 108 = -27 \]
\[ x = \frac{-b \pm \sqrt{D}}{2a} = \frac{-9 \pm \sqrt{-27}}{-6} \]
\[ x = \frac{3 + \sqrt{3}\, i}{2}\quad\text{or}\quad x = \frac{3 - \sqrt{3}\, i}{2} \]

The parabola y = -3x² + 9x - 9

xy
-1.5-29.25
-0.5-14.25
0.5-5.25
1.5-2.25
2.5-5.25
3.5-14.25
4.5-29.25

Questions

What are the solutions of -3x² + 9x - 9 = 0?

-3x² + 9x - 9 = 0 has no real solutions. The two complex solutions are x = (3 + √3 i)/2 and x = (3 - √3 i)/2.

Does -3x² + 9x - 9 = 0 factor?

Not over the integers, because the discriminant -27 is negative. Use the quadratic formula.

What is the vertex of y = -3x² + 9x - 9?

(3/2, -9/4). The parabola opens downwards, so this is the maximum.

What are the sum and product of the roots?

Sum = −b/a = 3, product = c/a = 3.

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