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Solve -x² - 4x + 12 = 0

The solutions of -x² - 4x + 12 = 0 are x = -6 or x = 2.

x² +x += 0
−x2−4x+12=0⇒x=−6, 2-x^2 - 4x + 12 = 0 \Rightarrow x = -6,\ 2
Discriminant D = 64 · x ≈ -6, 2

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

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Graph of y = -x² - 4x + 12-8-6-4-224-20-101020x = -6x = 2vertexy = -x² - 4x + 12
Graph of y = -x² - 4x + 12. The curve crosses the x-axis at x = -6 and x = 2; the vertex is at (-2, 16).

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 = -4, c = 12.

    −x2−4x+12=0-x^2 - 4x + 12 = 0
  2. 2

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

    D=(−4)2−4⋅(−1)⋅12=16+48=64D = (-4)^2 - 4\cdot(-1)\cdot12 = 16 + 48 = 64
  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+6)(x−2)=0-\left(x + 6\right)\left(x - 2\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+6=0⇒x=−6x−2=0⇒x=2x + 6 = 0 \Rightarrow x = -6\qquad x - 2 = 0 \Rightarrow x = 2
  6. 6

    Method 2, The quadratic formula works for every quadratic:

    x=−b±D2a=4±64−2x = \frac{-b \pm \sqrt{D}}{2a} = \frac{4 \pm \sqrt{64}}{-2}
  7. 7

    Simplify to get the solutions:

    x=−6orx=2x = -6\quad\text{or}\quad x = 2
  8. 8

    Check by putting x = -6 back in: a·x² + b·x + c = -1·(36) + (-4)·(-6) + 12 = 0 ✓

LaTeX source of the formulas on this page
\[ -x^2 - 4x + 12 = 0\ \Rightarrow\ x = -6,\ 2 \]
\[ -x^2 - 4x + 12 = 0 \]
\[ D = (-4)^2 - 4\cdot(-1)\cdot12 = 16 + 48 = 64 \]
\[ -\left(x + 6\right)\left(x - 2\right) = 0 \]
\[ x + 6 = 0 \Rightarrow x = -6\qquad x - 2 = 0 \Rightarrow x = 2 \]
\[ x = \frac{-b \pm \sqrt{D}}{2a} = \frac{4 \pm \sqrt{64}}{-2} \]
\[ x = -6\quad\text{or}\quad x = 2 \]

The parabola y = -x² - 4x + 12

xy
-57
-412
-315
-216
-115
012
17

Questions

What are the solutions of -x² - 4x + 12 = 0?

The solutions of -x² - 4x + 12 = 0 are x = -6 or x = 2.

Does -x² - 4x + 12 = 0 factor?

Yes: -1(x+6)(x-2) = 0.

What is the vertex of y = -x² - 4x + 12?

(-2, 16). The parabola opens downwards, so this is the maximum.

What are the sum and product of the roots?

Sum = −b/a = -4, product = c/a = -12.

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