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Solve -x² - 12x = 0

The solutions of -x² - 12x = 0 are x = -12 or x = 0.

x² +x += 0
−x2−12x=0⇒x=−12, 0-x^2 - 12x = 0 \Rightarrow x = -12,\ 0
Discriminant D = 144 · x ≈ -12, 0

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

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Graph of y = -x² - 12x-14-12-10-8-6-4-22-40-30-20-1010203040x = -12x = 0vertexy = -x² - 12x
Graph of y = -x² - 12x. The curve crosses the x-axis at x = -12 and x = 0; the vertex is at (-6, 36).

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 = -12, c = 0.

    −x2−12x=0-x^2 - 12x = 0
  2. 2

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

    D=(−12)2−4⋅(−1)⋅0=144+0=144D = (-12)^2 - 4\cdot(-1)\cdot0 = 144 + 0 = 144
  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+12)(x)=0-\left(x + 12\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+12=0⇒x=−12x=0⇒x=0x + 12 = 0 \Rightarrow x = -12\qquad x = 0 \Rightarrow x = 0
  6. 6

    Method 2, The quadratic formula works for every quadratic:

    x=−b±D2a=12±144−2x = \frac{-b \pm \sqrt{D}}{2a} = \frac{12 \pm \sqrt{144}}{-2}
  7. 7

    Simplify to get the solutions:

    x=−12orx=0x = -12\quad\text{or}\quad x = 0
  8. 8

    Check by putting x = -12 back in: a·x² + b·x + c = -1·(144) + (-12)·(-12) + 0 = 0 ✓

LaTeX source of the formulas on this page
\[ -x^2 - 12x = 0\ \Rightarrow\ x = -12,\ 0 \]
\[ -x^2 - 12x = 0 \]
\[ D = (-12)^2 - 4\cdot(-1)\cdot0 = 144 + 0 = 144 \]
\[ -\left(x + 12\right)\left(x\right) = 0 \]
\[ x + 12 = 0 \Rightarrow x = -12\qquad x = 0 \Rightarrow x = 0 \]
\[ x = \frac{-b \pm \sqrt{D}}{2a} = \frac{12 \pm \sqrt{144}}{-2} \]
\[ x = -12\quad\text{or}\quad x = 0 \]

The parabola y = -x² - 12x

xy
-927
-832
-735
-636
-535
-432
-327

Questions

What are the solutions of -x² - 12x = 0?

The solutions of -x² - 12x = 0 are x = -12 or x = 0.

Does -x² - 12x = 0 factor?

Yes: -1(x+12)(x) = 0.

What is the vertex of y = -x² - 12x?

(-6, 36). The parabola opens downwards, so this is the maximum.

What are the sum and product of the roots?

Sum = −b/a = -12, product = c/a = 0.

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