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Solve -x² - 5x = 0

The solutions of -x² - 5x = 0 are x = -5 or x = 0.

x² +x += 0
−x2−5x=0⇒x=−5, 0-x^2 - 5x = 0 \Rightarrow x = -5,\ 0
Discriminant D = 25 · x ≈ -5, 0

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

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Graph of y = -x² - 5x-7-6-5-4-3-2-112-20-15-10-55x = -5x = 0vertexy = -x² - 5x
Graph of y = -x² - 5x. The curve crosses the x-axis at x = -5 and x = 0; the vertex is at (-2.5, 6.25).

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 = -5, c = 0.

    −x2−5x=0-x^2 - 5x = 0
  2. 2

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

    D=(−5)2−4⋅(−1)⋅0=25+0=25D = (-5)^2 - 4\cdot(-1)\cdot0 = 25 + 0 = 25
  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+5)(x)=0-\left(x + 5\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+5=0⇒x=−5x=0⇒x=0x + 5 = 0 \Rightarrow x = -5\qquad x = 0 \Rightarrow x = 0
  6. 6

    Method 2, The quadratic formula works for every quadratic:

    x=−b±D2a=5±25−2x = \frac{-b \pm \sqrt{D}}{2a} = \frac{5 \pm \sqrt{25}}{-2}
  7. 7

    Simplify to get the solutions:

    x=−5orx=0x = -5\quad\text{or}\quad x = 0
  8. 8

    Check by putting x = -5 back in: a·x² + b·x + c = -1·(25) + (-5)·(-5) + 0 = 0 ✓

LaTeX source of the formulas on this page
\[ -x^2 - 5x = 0\ \Rightarrow\ x = -5,\ 0 \]
\[ -x^2 - 5x = 0 \]
\[ D = (-5)^2 - 4\cdot(-1)\cdot0 = 25 + 0 = 25 \]
\[ -\left(x + 5\right)\left(x\right) = 0 \]
\[ x + 5 = 0 \Rightarrow x = -5\qquad x = 0 \Rightarrow x = 0 \]
\[ x = \frac{-b \pm \sqrt{D}}{2a} = \frac{5 \pm \sqrt{25}}{-2} \]
\[ x = -5\quad\text{or}\quad x = 0 \]

The parabola y = -x² - 5x

xy
-5.5-2.75
-4.52.25
-3.55.25
-2.56.25
-1.55.25
-0.52.25
0.5-2.75

Questions

What are the solutions of -x² - 5x = 0?

The solutions of -x² - 5x = 0 are x = -5 or x = 0.

Does -x² - 5x = 0 factor?

Yes: -1(x+5)(x) = 0.

What is the vertex of y = -x² - 5x?

(-5/2, 25/4). The parabola opens downwards, so this is the maximum.

What are the sum and product of the roots?

Sum = −b/a = -5, product = c/a = 0.

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