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Solve -2x² - 9x - 8 = 0

The solutions of -2x² - 9x - 8 = 0 are x = (-9 + √17)/4 or x = (-9 - √17)/4, that is x ≈ -3.2807764 and x ≈ -1.2192236.

x² +x += 0
−2x2−9x−8=0⇒x=−9+174, −9−174-2x^2 - 9x - 8 = 0 \Rightarrow x = \frac{-9 + \sqrt{17}}{4},\ \frac{-9 - \sqrt{17}}{4}
Discriminant D = 17 · x ≈ -3.28078, -1.21922

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

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Graph of y = -2x² - 9x - 8-5-4-3-2-11-20-15-10-55x = -3.2807764x = -1.2192236vertexy = -2x² - 9x - 8
Graph of y = -2x² - 9x - 8. The curve crosses the x-axis at x = -3.2807764 and x = -1.2192236; the vertex is at (-2.25, 2.125).

How to work it out

  1. 1

    Write it in standard form a x² + b x + c = 0 and read off the coefficients: a = -2, b = -9, c = -8.

    −2x2−9x−8=0-2x^2 - 9x - 8 = 0
  2. 2

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

    D=(−9)2−4⋅(−2)⋅(−8)=81−64=17D = (-9)^2 - 4\cdot(-2)\cdot(-8) = 81 - 64 = 17
  3. 3

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

  4. 4

    The quadratic formula works for every quadratic:

    x=−b±D2a=9±17−4x = \frac{-b \pm \sqrt{D}}{2a} = \frac{9 \pm \sqrt{17}}{-4}
  5. 5

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

    x=−9+174orx=−9−174x = \frac{-9 + \sqrt{17}}{4}\quad\text{or}\quad x = \frac{-9 - \sqrt{17}}{4}
  6. 6

    As decimals: x ≈ -3.2807764 and x ≈ -1.2192236.

  7. 7

    Check numerically: for x ≈ -3.2807764 the left side is -3.5527137e-15 (zero up to rounding) ✓

LaTeX source of the formulas on this page
\[ -2x^2 - 9x - 8 = 0\ \Rightarrow\ x = \frac{-9 + \sqrt{17}}{4},\ \frac{-9 - \sqrt{17}}{4} \]
\[ -2x^2 - 9x - 8 = 0 \]
\[ D = (-9)^2 - 4\cdot(-2)\cdot(-8) = 81 - 64 = 17 \]
\[ x = \frac{-b \pm \sqrt{D}}{2a} = \frac{9 \pm \sqrt{17}}{-4} \]
\[ x = \frac{-9 + \sqrt{17}}{4}\quad\text{or}\quad x = \frac{-9 - \sqrt{17}}{4} \]

The parabola y = -2x² - 9x - 8

xy
-5.25-15.875
-4.25-5.875
-3.250.125
-2.252.125
-1.250.125
-0.25-5.875
0.75-15.875

Questions

What are the solutions of -2x² - 9x - 8 = 0?

The solutions of -2x² - 9x - 8 = 0 are x = (-9 + √17)/4 or x = (-9 - √17)/4, that is x ≈ -3.2807764 and x ≈ -1.2192236.

Does -2x² - 9x - 8 = 0 factor?

Not over the integers, because the discriminant 17 is not a perfect square. Use the quadratic formula.

What is the vertex of y = -2x² - 9x - 8?

(-9/4, 17/8). The parabola opens downwards, so this is the maximum.

What are the sum and product of the roots?

Sum = −b/a = -9/2, product = c/a = 4.

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