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Solve -2x² - 2x - 1 = 0

-2x² - 2x - 1 = 0 has no real solutions. The two complex solutions are x = (-1 + i)/2 and x = (-1 - i)/2.

x² +x += 0
−2x2−2x−1=0⇒x=−1+i2, −1−i2 (complex)-2x^2 - 2x - 1 = 0 \Rightarrow x = \frac{-1 + i}{2},\ \frac{-1 - i}{2}\ \text{(complex)}
Discriminant D = -4

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

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Graph of y = -2x² - 2x - 1-4-3-2-1123-30-25-20-15-10-5vertexy = -2x² - 2x - 1
Graph of y = -2x² - 2x - 1. The curve never reaches the x-axis, which is why there are no real solutions; the vertex is at (-0.5, -0.5).

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 = -2, c = -1.

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

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

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

    D is negative, so there are no real solutions; the two solutions are complex numbers.

  4. 4

    The quadratic formula works for every quadratic:

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

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

    x=−1+i2orx=−1−i2x = \frac{-1 + i}{2}\quad\text{or}\quad x = \frac{-1 - i}{2}
LaTeX source of the formulas on this page
\[ -2x^2 - 2x - 1 = 0\ \Rightarrow\ x = \frac{-1 + i}{2},\ \frac{-1 - i}{2} \]
\[ -2x^2 - 2x - 1 = 0 \]
\[ D = (-2)^2 - 4\cdot(-2)\cdot(-1) = 4 - 8 = -4 \]
\[ x = \frac{-b \pm \sqrt{D}}{2a} = \frac{2 \pm \sqrt{-4}}{-4} \]
\[ x = \frac{-1 + i}{2}\quad\text{or}\quad x = \frac{-1 - i}{2} \]

The parabola y = -2x² - 2x - 1

xy
-3.5-18.5
-2.5-8.5
-1.5-2.5
-0.5-0.5
0.5-2.5
1.5-8.5
2.5-18.5

Questions

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

-2x² - 2x - 1 = 0 has no real solutions. The two complex solutions are x = (-1 + i)/2 and x = (-1 - i)/2.

Does -2x² - 2x - 1 = 0 factor?

Not over the integers, because the discriminant -4 is negative. Use the quadratic formula.

What is the vertex of y = -2x² - 2x - 1?

(-1/2, -1/2). The parabola opens downwards, so this is the maximum.

What are the sum and product of the roots?

Sum = −b/a = -1, product = c/a = 1/2.

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