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Solve -2x² - 5x - 1 = 0

The solutions of -2x² - 5x - 1 = 0 are x = (-5 + √17)/4 or x = (-5 - √17)/4, that is x ≈ -2.2807764 and x ≈ -0.21922359.

x² +x += 0
−2x2−5x−1=0⇒x=−5+174, −5−174-2x^2 - 5x - 1 = 0 \Rightarrow x = \frac{-5 + \sqrt{17}}{4},\ \frac{-5 - \sqrt{17}}{4}
Discriminant D = 17 · x ≈ -2.28078, -0.219224

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

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Graph of y = -2x² - 5x - 1-4-3-2-112-20-15-10-55x = -2.2807764x = -0.21922359vertexy = -2x² - 5x - 1
Graph of y = -2x² - 5x - 1. The curve crosses the x-axis at x = -2.2807764 and x = -0.21922359; the vertex is at (-1.25, 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 = -5, c = -1.

    −2x2−5x−1=0-2x^2 - 5x - 1 = 0
  2. 2

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

    D=(−5)2−4⋅(−2)⋅(−1)=25−8=17D = (-5)^2 - 4\cdot(-2)\cdot(-1) = 25 - 8 = 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=5±17−4x = \frac{-b \pm \sqrt{D}}{2a} = \frac{5 \pm \sqrt{17}}{-4}
  5. 5

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

    x=−5+174orx=−5−174x = \frac{-5 + \sqrt{17}}{4}\quad\text{or}\quad x = \frac{-5 - \sqrt{17}}{4}
  6. 6

    As decimals: x ≈ -2.2807764 and x ≈ -0.21922359.

  7. 7

    Check numerically: for x ≈ -2.2807764 the left side is -1.7763568e-15 (zero up to rounding) ✓

LaTeX source of the formulas on this page
\[ -2x^2 - 5x - 1 = 0\ \Rightarrow\ x = \frac{-5 + \sqrt{17}}{4},\ \frac{-5 - \sqrt{17}}{4} \]
\[ -2x^2 - 5x - 1 = 0 \]
\[ D = (-5)^2 - 4\cdot(-2)\cdot(-1) = 25 - 8 = 17 \]
\[ x = \frac{-b \pm \sqrt{D}}{2a} = \frac{5 \pm \sqrt{17}}{-4} \]
\[ x = \frac{-5 + \sqrt{17}}{4}\quad\text{or}\quad x = \frac{-5 - \sqrt{17}}{4} \]

The parabola y = -2x² - 5x - 1

xy
-4.25-15.875
-3.25-5.875
-2.250.125
-1.252.125
-0.250.125
0.75-5.875
1.75-15.875

Questions

What are the solutions of -2x² - 5x - 1 = 0?

The solutions of -2x² - 5x - 1 = 0 are x = (-5 + √17)/4 or x = (-5 - √17)/4, that is x ≈ -2.2807764 and x ≈ -0.21922359.

Does -2x² - 5x - 1 = 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 - 5x - 1?

(-5/4, 17/8). The parabola opens downwards, so this is the maximum.

What are the sum and product of the roots?

Sum = −b/a = -5/2, product = c/a = 1/2.

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    Solve -2x² - 5x - 1 = 0: x = (-5 + √17)/4 or x = (-5 - √17)/4 (steps) | WordGPT