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Solve -3x² + 2x = 0

The solutions of -3x² + 2x = 0 are x = 0 or x = 2/3.

x² +x += 0
−3x2+2x=0⇒x=0, 23-3x^2 + 2x = 0 \Rightarrow x = 0,\ \frac{2}{3}
Discriminant D = 4 · x ≈ 0, 0.666667

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

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Graph of y = -3x² + 2x-2-1123-25-20-15-10-5x = 0x = 0.66666667vertexy = -3x² + 2x
Graph of y = -3x² + 2x. The curve crosses the x-axis at x = 0 and x = 0.66666667; the vertex is at (0.33333333, 0.33333333).

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 = 2, c = 0.

    −3x2+2x=0-3x^2 + 2x = 0
  2. 2

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

    D=22−4⋅(−3)⋅0=4+0=4D = 2^2 - 4\cdot(-3)\cdot0 = 4 + 0 = 4
  3. 3

    D is positive, so there are two different real solutions.

  4. 4

    Method 1, factoring. The roots are rational, so the left side factors:

    −(x)(3x−2)=0-\left(x\right)\left(3x - 2\right) = 0
  5. 5

    A product is zero when one of its factors is zero, so set each factor equal to 0 and solve:

    x=0⇒x=03x−2=0⇒x=23x = 0 \Rightarrow x = 0\qquad 3x - 2 = 0 \Rightarrow x = \frac{2}{3}
  6. 6

    Method 2, The quadratic formula works for every quadratic:

    x=−b±D2a=−2±4−6x = \frac{-b \pm \sqrt{D}}{2a} = \frac{-2 \pm \sqrt{4}}{-6}
  7. 7

    Simplify to get the solutions:

    x=0orx=23x = 0\quad\text{or}\quad x = \frac{2}{3}
  8. 8

    Check by putting x = 0 back in: a·x² + b·x + c = -3·(0) + 2·(0) + 0 = 0 ✓

LaTeX source of the formulas on this page
\[ -3x^2 + 2x = 0\ \Rightarrow\ x = 0,\ \frac{2}{3} \]
\[ -3x^2 + 2x = 0 \]
\[ D = 2^2 - 4\cdot(-3)\cdot0 = 4 + 0 = 4 \]
\[ -\left(x\right)\left(3x - 2\right) = 0 \]
\[ x = 0 \Rightarrow x = 0\qquad 3x - 2 = 0 \Rightarrow x = \frac{2}{3} \]
\[ x = \frac{-b \pm \sqrt{D}}{2a} = \frac{-2 \pm \sqrt{4}}{-6} \]
\[ x = 0\quad\text{or}\quad x = \frac{2}{3} \]

The parabola y = -3x² + 2x

xy
-2.6667-26.667267
-1.6667-11.667067
-0.6667-2.6668667
0.33330.33333333
1.3333-2.6664667
2.3333-11.666267
3.3333-26.666067

Questions

What are the solutions of -3x² + 2x = 0?

The solutions of -3x² + 2x = 0 are x = 0 or x = 2/3.

Does -3x² + 2x = 0 factor?

Yes: -1(x)(3x-2) = 0.

What is the vertex of y = -3x² + 2x?

(1/3, 1/3). The parabola opens downwards, so this is the maximum.

What are the sum and product of the roots?

Sum = −b/a = 2/3, product = c/a = 0.

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