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Solve -2x² - 2x - 3 = 0

-2x² - 2x - 3 = 0 has no real solutions. The two complex solutions are x = (-1 + √5 i)/2 and x = (-1 - √5 i)/2.

x² +x += 0
−2x2−2x−3=0⇒x=−1+5 i2, −1−5 i2 (complex)-2x^2 - 2x - 3 = 0 \Rightarrow x = \frac{-1 + \sqrt{5}\, i}{2},\ \frac{-1 - \sqrt{5}\, i}{2}\ \text{(complex)}
Discriminant D = -20

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

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Graph of y = -2x² - 2x - 3-4-3-2-1123-35-30-25-20-15-10-5vertexy = -2x² - 2x - 3
Graph of y = -2x² - 2x - 3. The curve never reaches the x-axis, which is why there are no real solutions; the vertex is at (-0.5, -2.5).

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 = -3.

    −2x2−2x−3=0-2x^2 - 2x - 3 = 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)⋅(−3)=4−24=−20D = (-2)^2 - 4\cdot(-2)\cdot(-3) = 4 - 24 = -20
  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±−20−4x = \frac{-b \pm \sqrt{D}}{2a} = \frac{2 \pm \sqrt{-20}}{-4}
  5. 5

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

    x=−1+5 i2orx=−1−5 i2x = \frac{-1 + \sqrt{5}\, i}{2}\quad\text{or}\quad x = \frac{-1 - \sqrt{5}\, i}{2}
LaTeX source of the formulas on this page
\[ -2x^2 - 2x - 3 = 0\ \Rightarrow\ x = \frac{-1 + \sqrt{5}\, i}{2},\ \frac{-1 - \sqrt{5}\, i}{2} \]
\[ -2x^2 - 2x - 3 = 0 \]
\[ D = (-2)^2 - 4\cdot(-2)\cdot(-3) = 4 - 24 = -20 \]
\[ x = \frac{-b \pm \sqrt{D}}{2a} = \frac{2 \pm \sqrt{-20}}{-4} \]
\[ x = \frac{-1 + \sqrt{5}\, i}{2}\quad\text{or}\quad x = \frac{-1 - \sqrt{5}\, i}{2} \]

The parabola y = -2x² - 2x - 3

xy
-3.5-20.5
-2.5-10.5
-1.5-4.5
-0.5-2.5
0.5-4.5
1.5-10.5
2.5-20.5

Questions

What are the solutions of -2x² - 2x - 3 = 0?

-2x² - 2x - 3 = 0 has no real solutions. The two complex solutions are x = (-1 + √5 i)/2 and x = (-1 - √5 i)/2.

Does -2x² - 2x - 3 = 0 factor?

Not over the integers, because the discriminant -20 is negative. Use the quadratic formula.

What is the vertex of y = -2x² - 2x - 3?

(-1/2, -5/2). 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 = 3/2.

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