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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, −133x^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-2-1125101520x = 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=03x^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\cdot3\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±16x = \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
\[ 3x^2 + x = 0\ \Rightarrow\ x = 0,\ -\frac{1}{3} \]
\[ 3x^2 + x = 0 \]
\[ D = 1^2 - 4\cdot3\cdot0 = 1 + 0 = 1 \]
\[ \left(x\right)\left(3x + 1\right) = 0 \]
\[ x = 0 \Rightarrow x = 0\qquad 3x + 1 = 0 \Rightarrow x = -\frac{1}{3} \]
\[ x = \frac{-b \pm \sqrt{D}}{2a} = \frac{-1 \pm \sqrt{1}}{6} \]
\[ x = 0\quad\text{or}\quad x = -\frac{1}{3} \]

The parabola y = 3x² + x

xy
-3.166726.917267
-2.166711.917067
-1.16672.9168667
-0.1667-0.08333333
0.83332.9164667
1.833311.916267
2.833326.916067

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: (x)(3x+1) = 0.

What is the vertex of y = 3x² + x?

(-1/6, -1/12). The parabola opens upwards, so this is the minimum.

What are the sum and product of the roots?

Sum = −b/a = -1/3, product = c/a = 0.

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