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Solve -2x² - 3x - 12 = 0

-2x² - 3x - 12 = 0 has no real solutions. The two complex solutions are x = (-3 + √87 i)/4 and x = (-3 - √87 i)/4.

x² +x += 0
−2x2−3x−12=0⇒x=−3+87 i4, −3−87 i4 (complex)-2x^2 - 3x - 12 = 0 \Rightarrow x = \frac{-3 + \sqrt{87}\, i}{4},\ \frac{-3 - \sqrt{87}\, i}{4}\ \text{(complex)}
Discriminant D = -87

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

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Graph of y = -2x² - 3x - 12-4-3-2-1123-45-40-35-30-25-20-15-10vertexy = -2x² - 3x - 12
Graph of y = -2x² - 3x - 12. The curve never reaches the x-axis, which is why there are no real solutions; the vertex is at (-0.75, -10.875).

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 = -12.

    −2x2−3x−12=0-2x^2 - 3x - 12 = 0
  2. 2

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

    D=(−3)2−4⋅(−2)⋅(−12)=9−96=−87D = (-3)^2 - 4\cdot(-2)\cdot(-12) = 9 - 96 = -87
  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±−87−4x = \frac{-b \pm \sqrt{D}}{2a} = \frac{3 \pm \sqrt{-87}}{-4}
  5. 5

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

    x=−3+87 i4orx=−3−87 i4x = \frac{-3 + \sqrt{87}\, i}{4}\quad\text{or}\quad x = \frac{-3 - \sqrt{87}\, i}{4}
LaTeX source of the formulas on this page
\[ -2x^2 - 3x - 12 = 0\ \Rightarrow\ x = \frac{-3 + \sqrt{87}\, i}{4},\ \frac{-3 - \sqrt{87}\, i}{4} \]
\[ -2x^2 - 3x - 12 = 0 \]
\[ D = (-3)^2 - 4\cdot(-2)\cdot(-12) = 9 - 96 = -87 \]
\[ x = \frac{-b \pm \sqrt{D}}{2a} = \frac{3 \pm \sqrt{-87}}{-4} \]
\[ x = \frac{-3 + \sqrt{87}\, i}{4}\quad\text{or}\quad x = \frac{-3 - \sqrt{87}\, i}{4} \]

The parabola y = -2x² - 3x - 12

xy
-3.75-28.875
-2.75-18.875
-1.75-12.875
-0.75-10.875
0.25-12.875
1.25-18.875
2.25-28.875

Questions

What are the solutions of -2x² - 3x - 12 = 0?

-2x² - 3x - 12 = 0 has no real solutions. The two complex solutions are x = (-3 + √87 i)/4 and x = (-3 - √87 i)/4.

Does -2x² - 3x - 12 = 0 factor?

Not over the integers, because the discriminant -87 is negative. Use the quadratic formula.

What is the vertex of y = -2x² - 3x - 12?

(-3/4, -87/8). 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 = 6.

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