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Solve -2x² + 3x - 8 = 0

-2x² + 3x - 8 = 0 has no real solutions. The two complex solutions are x = (3 + √55 i)/4 and x = (3 - √55 i)/4.

x² +x += 0
−2x2+3x−8=0⇒x=3+55 i4, 3−55 i4 (complex)-2x^2 + 3x - 8 = 0 \Rightarrow x = \frac{3 + \sqrt{55}\, i}{4},\ \frac{3 - \sqrt{55}\, i}{4}\ \text{(complex)}
Discriminant D = -55

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

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Graph of y = -2x² + 3x - 8-3-2-11234-40-35-30-25-20-15-10-5vertexy = -2x² + 3x - 8
Graph of y = -2x² + 3x - 8. The curve never reaches the x-axis, which is why there are no real solutions; the vertex is at (0.75, -6.875).

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 = 3, c = -8.

    −2x2+3x−8=0-2x^2 + 3x - 8 = 0
  2. 2

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

    D=32−4⋅(−2)⋅(−8)=9−64=−55D = 3^2 - 4\cdot(-2)\cdot(-8) = 9 - 64 = -55
  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=−3±−55−4x = \frac{-b \pm \sqrt{D}}{2a} = \frac{-3 \pm \sqrt{-55}}{-4}
  5. 5

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

    x=3+55 i4orx=3−55 i4x = \frac{3 + \sqrt{55}\, i}{4}\quad\text{or}\quad x = \frac{3 - \sqrt{55}\, i}{4}
LaTeX source of the formulas on this page
\[ -2x^2 + 3x - 8 = 0\ \Rightarrow\ x = \frac{3 + \sqrt{55}\, i}{4},\ \frac{3 - \sqrt{55}\, i}{4} \]
\[ -2x^2 + 3x - 8 = 0 \]
\[ D = 3^2 - 4\cdot(-2)\cdot(-8) = 9 - 64 = -55 \]
\[ x = \frac{-b \pm \sqrt{D}}{2a} = \frac{-3 \pm \sqrt{-55}}{-4} \]
\[ x = \frac{3 + \sqrt{55}\, i}{4}\quad\text{or}\quad x = \frac{3 - \sqrt{55}\, i}{4} \]

The parabola y = -2x² + 3x - 8

xy
-2.25-24.875
-1.25-14.875
-0.25-8.875
0.75-6.875
1.75-8.875
2.75-14.875
3.75-24.875

Questions

What are the solutions of -2x² + 3x - 8 = 0?

-2x² + 3x - 8 = 0 has no real solutions. The two complex solutions are x = (3 + √55 i)/4 and x = (3 - √55 i)/4.

Does -2x² + 3x - 8 = 0 factor?

Not over the integers, because the discriminant -55 is negative. Use the quadratic formula.

What is the vertex of y = -2x² + 3x - 8?

(3/4, -55/8). The parabola opens downwards, so this is the maximum.

What are the sum and product of the roots?

Sum = −b/a = 3/2, product = c/a = 4.

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