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Solve -3x² - x = 0

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

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

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 = -0.33333333x = 0vertexy = -3x² - x
Graph of y = -3x² - x. The curve crosses the x-axis at x = -0.33333333 and x = 0; 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=(−1)2−4⋅(−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:

    −(3x+1)(x)=0-\left(3x + 1\right)\left(x\right) = 0
  5. 5

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

    3x+1=0⇒x=−13x=0⇒x=03x + 1 = 0 \Rightarrow x = -\frac{1}{3}\qquad x = 0 \Rightarrow x = 0
  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=−13orx=0x = -\frac{1}{3}\quad\text{or}\quad x = 0
  8. 8

    Check by putting x = -1/3 back in: a·x² + b·x + c = -3·(1/9) + (-1)·(-1/3) + 0 = 0 ✓

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

The parabola y = -3x² - x

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

Questions

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

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

Does -3x² - x = 0 factor?

Yes: -1(3x+1)(x) = 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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