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Solve -2x² + x - 8 = 0

-2x² + x - 8 = 0 has no real solutions. The two complex solutions are x = (1 + 3√7 i)/4 and x = (1 - 3√7 i)/4.

x² +x += 0
−2x2+x−8=0⇒x=1+37 i4, 1−37 i4 (complex)-2x^2 + x - 8 = 0 \Rightarrow x = \frac{1 + 3\sqrt{7}\, i}{4},\ \frac{1 - 3\sqrt{7}\, i}{4}\ \text{(complex)}
Discriminant D = -63

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

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Graph of y = -2x² + x - 8-3-2-11234-40-35-30-25-20-15-10-5vertexy = -2x² + x - 8
Graph of y = -2x² + x - 8. The curve never reaches the x-axis, which is why there are no real solutions; the vertex is at (0.25, -7.875).

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 = 1, c = -8.

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

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

    D=12−4⋅(−2)⋅(−8)=1−64=−63D = 1^2 - 4\cdot(-2)\cdot(-8) = 1 - 64 = -63
  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=−1±−63−4x = \frac{-b \pm \sqrt{D}}{2a} = \frac{-1 \pm \sqrt{-63}}{-4}
  5. 5

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

    x=1+37 i4orx=1−37 i4x = \frac{1 + 3\sqrt{7}\, i}{4}\quad\text{or}\quad x = \frac{1 - 3\sqrt{7}\, i}{4}
LaTeX source of the formulas on this page
\[ -2x^2 + x - 8 = 0\ \Rightarrow\ x = \frac{1 + 3\sqrt{7}\, i}{4},\ \frac{1 - 3\sqrt{7}\, i}{4} \]
\[ -2x^2 + x - 8 = 0 \]
\[ D = 1^2 - 4\cdot(-2)\cdot(-8) = 1 - 64 = -63 \]
\[ x = \frac{-b \pm \sqrt{D}}{2a} = \frac{-1 \pm \sqrt{-63}}{-4} \]
\[ x = \frac{1 + 3\sqrt{7}\, i}{4}\quad\text{or}\quad x = \frac{1 - 3\sqrt{7}\, i}{4} \]

The parabola y = -2x² + x - 8

xy
-2.75-25.875
-1.75-15.875
-0.75-9.875
0.25-7.875
1.25-9.875
2.25-15.875
3.25-25.875

Questions

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

-2x² + x - 8 = 0 has no real solutions. The two complex solutions are x = (1 + 3√7 i)/4 and x = (1 - 3√7 i)/4.

Does -2x² + x - 8 = 0 factor?

Not over the integers, because the discriminant -63 is negative. Use the quadratic formula.

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

(1/4, -63/8). The parabola opens downwards, so this is the maximum.

What are the sum and product of the roots?

Sum = −b/a = 1/2, product = c/a = 4.

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