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Solve -x² - x = 0

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

x² +x += 0
−x2−x=0⇒x=−1, 0-x^2 - x = 0 \Rightarrow x = -1,\ 0
Discriminant D = 1 · x ≈ -1, 0

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

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Graph of y = -x² - x-3-2-112-8-6-4-2x = -1x = 0vertexy = -x² - x
Graph of y = -x² - x. The curve crosses the x-axis at x = -1 and x = 0; the vertex is at (-0.5, 0.25).

How to work it out

  1. 1

    Write it in standard form a x² + b x + c = 0 and read off the coefficients: a = -1, b = -1, c = 0.

    −x2−x=0-x^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⋅(−1)⋅0=1+0=1D = (-1)^2 - 4\cdot(-1)\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+1)(x)=0-\left(x + 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:

    x+1=0⇒x=−1x=0⇒x=0x + 1 = 0 \Rightarrow x = -1\qquad x = 0 \Rightarrow x = 0
  6. 6

    Method 2, The quadratic formula works for every quadratic:

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

    Simplify to get the solutions:

    x=−1orx=0x = -1\quad\text{or}\quad x = 0
  8. 8

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

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

The parabola y = -x² - x

xy
-3.5-8.75
-2.5-3.75
-1.5-0.75
-0.50.25
0.5-0.75
1.5-3.75
2.5-8.75

Questions

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

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

Does -x² - x = 0 factor?

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

What is the vertex of y = -x² - x?

(-1/2, 1/4). The parabola opens downwards, so this is the maximum.

What are the sum and product of the roots?

Sum = −b/a = -1, product = c/a = 0.

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