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Solve -3x² + x = 0

The solutions of -3x² + x = 0 are x = 0 or x = 1/3.

x² +x += 0
−3x2+x=0⇒x=0, 13-3x^2 + x = 0 \Rightarrow x = 0,\ \frac{1}{3}
Discriminant D = 1 · x ≈ 0, 0.333333

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

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Graph of y = -3x² + x-2-112-20-15-10-5x = 0x = 0.33333333vertexy = -3x² + x
Graph of y = -3x² + x. The curve crosses the x-axis at x = 0 and x = 0.33333333; the vertex is at (0.16666667, 0.083333333).

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 = 1, c = 0.

    −3x2+x=0-3x^2 + x = 0
  2. 2

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

    D=12−4⋅(−3)⋅0=1+0=1D = 1^2 - 4\cdot(-3)\cdot0 = 1 + 0 = 1
  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−1)=0-\left(x\right)\left(3x - 1\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−1=0⇒x=13x = 0 \Rightarrow x = 0\qquad 3x - 1 = 0 \Rightarrow x = \frac{1}{3}
  6. 6

    Method 2, The quadratic formula works for every quadratic:

    x=−b±D2a=−1±1−6x = \frac{-b \pm \sqrt{D}}{2a} = \frac{-1 \pm \sqrt{1}}{-6}
  7. 7

    Simplify to get the solutions:

    x=0orx=13x = 0\quad\text{or}\quad x = \frac{1}{3}
  8. 8

    Check by putting x = 0 back in: a·x² + b·x + c = -3·(0) + 1·(0) + 0 = 0 ✓

LaTeX source of the formulas on this page
\[ -3x^2 + x = 0\ \Rightarrow\ x = 0,\ \frac{1}{3} \]
\[ -3x^2 + x = 0 \]
\[ D = 1^2 - 4\cdot(-3)\cdot0 = 1 + 0 = 1 \]
\[ -\left(x\right)\left(3x - 1\right) = 0 \]
\[ x = 0 \Rightarrow x = 0\qquad 3x - 1 = 0 \Rightarrow x = \frac{1}{3} \]
\[ x = \frac{-b \pm \sqrt{D}}{2a} = \frac{-1 \pm \sqrt{1}}{-6} \]
\[ x = 0\quad\text{or}\quad x = \frac{1}{3} \]

The parabola y = -3x² + x

xy
-2.8333-26.916067
-1.8333-11.916267
-0.8333-2.9164667
0.16670.08333333
1.1667-2.9168667
2.1667-11.917067
3.1667-26.917267

Questions

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

The solutions of -3x² + x = 0 are x = 0 or x = 1/3.

Does -3x² + x = 0 factor?

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

What is the vertex of y = -3x² + x?

(1/6, 1/12). The parabola opens downwards, so this is the maximum.

What are the sum and product of the roots?

Sum = −b/a = 1/3, product = c/a = 0.

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