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Solve -x² - 12x - 8 = 0

The solutions of -x² - 12x - 8 = 0 are x = -6 + 2√7 or x = -6 - 2√7, that is x ≈ -11.291503 and x ≈ -0.70849738.

x² +x += 0
−x2−12x−8=0⇒x=−6+27, −6−27-x^2 - 12x - 8 = 0 \Rightarrow x = -6 + 2\sqrt{7},\ -6 - 2\sqrt{7}
Discriminant D = 112 · x ≈ -11.2915, -0.708497

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

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Graph of y = -x² - 12x - 8-12-10-8-6-4-2-30-20-10102030x = -11.291503x = -0.70849738vertexy = -x² - 12x - 8
Graph of y = -x² - 12x - 8. The curve crosses the x-axis at x = -11.291503 and x = -0.70849738; the vertex is at (-6, 28).

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 = -8.

    −x2−12x−8=0-x^2 - 12x - 8 = 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)⋅(−8)=144−32=112D = (-12)^2 - 4\cdot(-1)\cdot(-8) = 144 - 32 = 112
  3. 3

    D is positive, so there are two different real solutions.

  4. 4

    The quadratic formula works for every quadratic:

    x=−b±D2a=12±112−2x = \frac{-b \pm \sqrt{D}}{2a} = \frac{12 \pm \sqrt{112}}{-2}
  5. 5

    Simplify (pull the square out of the root) to get the solutions:

    x=−6+27orx=−6−27x = -6 + 2\sqrt{7}\quad\text{or}\quad x = -6 - 2\sqrt{7}
  6. 6

    As decimals: x ≈ -11.291503 and x ≈ -0.70849738.

  7. 7

    Check numerically: for x ≈ -11.291503 the left side is 0 (zero up to rounding) ✓

LaTeX source of the formulas on this page
\[ -x^2 - 12x - 8 = 0\ \Rightarrow\ x = -6 + 2\sqrt{7},\ -6 - 2\sqrt{7} \]
\[ -x^2 - 12x - 8 = 0 \]
\[ D = (-12)^2 - 4\cdot(-1)\cdot(-8) = 144 - 32 = 112 \]
\[ x = \frac{-b \pm \sqrt{D}}{2a} = \frac{12 \pm \sqrt{112}}{-2} \]
\[ x = -6 + 2\sqrt{7}\quad\text{or}\quad x = -6 - 2\sqrt{7} \]

The parabola y = -x² - 12x - 8

xy
-919
-824
-727
-628
-527
-424
-319

Questions

What are the solutions of -x² - 12x - 8 = 0?

The solutions of -x² - 12x - 8 = 0 are x = -6 + 2√7 or x = -6 - 2√7, that is x ≈ -11.291503 and x ≈ -0.70849738.

Does -x² - 12x - 8 = 0 factor?

Not over the integers, because the discriminant 112 is not a perfect square. Use the quadratic formula.

What is the vertex of y = -x² - 12x - 8?

(-6, 28). 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 = 8.

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    Solve -x² - 12x - 8 = 0: x = -6 + 2√7 or x = -6 - 2√7 (steps) | WordGPT