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Solve -3x² + 4x - 3 = 0

-3x² + 4x - 3 = 0 has no real solutions. The two complex solutions are x = (2 + √5 i)/3 and x = (2 - √5 i)/3.

x² +x += 0
−3x2+4x−3=0⇒x=2+5 i3, 2−5 i3 (complex)-3x^2 + 4x - 3 = 0 \Rightarrow x = \frac{2 + \sqrt{5}\, i}{3},\ \frac{2 - \sqrt{5}\, i}{3}\ \text{(complex)}
Discriminant D = -20

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

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Graph of y = -3x² + 4x - 3-3-2-11234-50-40-30-20-10vertexy = -3x² + 4x - 3
Graph of y = -3x² + 4x - 3. The curve never reaches the x-axis, which is why there are no real solutions; the vertex is at (0.66666667, -1.6666667).

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 = -3.

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

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

    D=42−4⋅(−3)⋅(−3)=16−36=−20D = 4^2 - 4\cdot(-3)\cdot(-3) = 16 - 36 = -20
  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±−20−6x = \frac{-b \pm \sqrt{D}}{2a} = \frac{-4 \pm \sqrt{-20}}{-6}
  5. 5

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

    x=2+5 i3orx=2−5 i3x = \frac{2 + \sqrt{5}\, i}{3}\quad\text{or}\quad x = \frac{2 - \sqrt{5}\, i}{3}
LaTeX source of the formulas on this page
\[ -3x^2 + 4x - 3 = 0\ \Rightarrow\ x = \frac{2 + \sqrt{5}\, i}{3},\ \frac{2 - \sqrt{5}\, i}{3} \]
\[ -3x^2 + 4x - 3 = 0 \]
\[ D = 4^2 - 4\cdot(-3)\cdot(-3) = 16 - 36 = -20 \]
\[ x = \frac{-b \pm \sqrt{D}}{2a} = \frac{-4 \pm \sqrt{-20}}{-6} \]
\[ x = \frac{2 + \sqrt{5}\, i}{3}\quad\text{or}\quad x = \frac{2 - \sqrt{5}\, i}{3} \]

The parabola y = -3x² + 4x - 3

xy
-2.3333-28.666067
-1.3333-13.666267
-0.3333-4.6664667
0.6667-1.6666667
1.6667-4.6668667
2.6667-13.667067
3.6667-28.667267

Questions

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

-3x² + 4x - 3 = 0 has no real solutions. The two complex solutions are x = (2 + √5 i)/3 and x = (2 - √5 i)/3.

Does -3x² + 4x - 3 = 0 factor?

Not over the integers, because the discriminant -20 is negative. Use the quadratic formula.

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

(2/3, -5/3). 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 = 1.

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