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Solve -x² - 8x - 5 = 0

The solutions of -x² - 8x - 5 = 0 are x = -4 + √11 or x = -4 - √11, that is x ≈ -7.3166248 and x ≈ -0.68337521.

x² +x += 0
−x2−8x−5=0⇒x=−4+11, −4−11-x^2 - 8x - 5 = 0 \Rightarrow x = -4 + \sqrt{11},\ -4 - \sqrt{11}
Discriminant D = 44 · x ≈ -7.31662, -0.683375

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

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Graph of y = -x² - 8x - 5-8-6-4-2-25-20-15-10-551015x = -7.3166248x = -0.68337521vertexy = -x² - 8x - 5
Graph of y = -x² - 8x - 5. The curve crosses the x-axis at x = -7.3166248 and x = -0.68337521; the vertex is at (-4, 11).

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 = -8, c = -5.

    −x2−8x−5=0-x^2 - 8x - 5 = 0
  2. 2

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

    D=(−8)2−4⋅(−1)⋅(−5)=64−20=44D = (-8)^2 - 4\cdot(-1)\cdot(-5) = 64 - 20 = 44
  3. 3

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

  4. 4

    The quadratic formula works for every quadratic:

    x=−b±D2a=8±44−2x = \frac{-b \pm \sqrt{D}}{2a} = \frac{8 \pm \sqrt{44}}{-2}
  5. 5

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

    x=−4+11orx=−4−11x = -4 + \sqrt{11}\quad\text{or}\quad x = -4 - \sqrt{11}
  6. 6

    As decimals: x ≈ -7.3166248 and x ≈ -0.68337521.

  7. 7

    Check numerically: for x ≈ -7.3166248 the left side is 0 (zero up to rounding) ✓

LaTeX source of the formulas on this page
\[ -x^2 - 8x - 5 = 0\ \Rightarrow\ x = -4 + \sqrt{11},\ -4 - \sqrt{11} \]
\[ -x^2 - 8x - 5 = 0 \]
\[ D = (-8)^2 - 4\cdot(-1)\cdot(-5) = 64 - 20 = 44 \]
\[ x = \frac{-b \pm \sqrt{D}}{2a} = \frac{8 \pm \sqrt{44}}{-2} \]
\[ x = -4 + \sqrt{11}\quad\text{or}\quad x = -4 - \sqrt{11} \]

The parabola y = -x² - 8x - 5

xy
-72
-67
-510
-411
-310
-27
-12

Questions

What are the solutions of -x² - 8x - 5 = 0?

The solutions of -x² - 8x - 5 = 0 are x = -4 + √11 or x = -4 - √11, that is x ≈ -7.3166248 and x ≈ -0.68337521.

Does -x² - 8x - 5 = 0 factor?

Not over the integers, because the discriminant 44 is not a perfect square. Use the quadratic formula.

What is the vertex of y = -x² - 8x - 5?

(-4, 11). The parabola opens downwards, so this is the maximum.

What are the sum and product of the roots?

Sum = −b/a = -8, product = c/a = 5.

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    Solve -x² - 8x - 5 = 0: x = -4 + √11 or x = -4 - √11 (steps) | WordGPT