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Solve -3x² - 8x - 10 = 0

-3x² - 8x - 10 = 0 has no real solutions. The two complex solutions are x = (-4 + √14 i)/3 and x = (-4 - √14 i)/3.

x² +x += 0
−3x2−8x−10=0⇒x=−4+14 i3, −4−14 i3 (complex)-3x^2 - 8x - 10 = 0 \Rightarrow x = \frac{-4 + \sqrt{14}\, i}{3},\ \frac{-4 - \sqrt{14}\, i}{3}\ \text{(complex)}
Discriminant D = -56

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

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Graph of y = -3x² - 8x - 10-5-4-3-2-112-50-40-30-20-10vertexy = -3x² - 8x - 10
Graph of y = -3x² - 8x - 10. The curve never reaches the x-axis, which is why there are no real solutions; the vertex is at (-1.3333333, -4.6666667).

How to work it out

  1. 1

    Write it in standard form a x² + b x + c = 0 and read off the coefficients: a = -3, b = -8, c = -10.

    −3x2−8x−10=0-3x^2 - 8x - 10 = 0
  2. 2

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

    D=(−8)2−4⋅(−3)⋅(−10)=64−120=−56D = (-8)^2 - 4\cdot(-3)\cdot(-10) = 64 - 120 = -56
  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=8±−56−6x = \frac{-b \pm \sqrt{D}}{2a} = \frac{8 \pm \sqrt{-56}}{-6}
  5. 5

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

    x=−4+14 i3orx=−4−14 i3x = \frac{-4 + \sqrt{14}\, i}{3}\quad\text{or}\quad x = \frac{-4 - \sqrt{14}\, i}{3}
LaTeX source of the formulas on this page
\[ -3x^2 - 8x - 10 = 0\ \Rightarrow\ x = \frac{-4 + \sqrt{14}\, i}{3},\ \frac{-4 - \sqrt{14}\, i}{3} \]
\[ -3x^2 - 8x - 10 = 0 \]
\[ D = (-8)^2 - 4\cdot(-3)\cdot(-10) = 64 - 120 = -56 \]
\[ x = \frac{-b \pm \sqrt{D}}{2a} = \frac{8 \pm \sqrt{-56}}{-6} \]
\[ x = \frac{-4 + \sqrt{14}\, i}{3}\quad\text{or}\quad x = \frac{-4 - \sqrt{14}\, i}{3} \]

The parabola y = -3x² - 8x - 10

xy
-4.3333-31.666067
-3.3333-16.666267
-2.3333-7.6664667
-1.3333-4.6666667
-0.3333-7.6668667
0.6667-16.667067
1.6667-31.667267

Questions

What are the solutions of -3x² - 8x - 10 = 0?

-3x² - 8x - 10 = 0 has no real solutions. The two complex solutions are x = (-4 + √14 i)/3 and x = (-4 - √14 i)/3.

Does -3x² - 8x - 10 = 0 factor?

Not over the integers, because the discriminant -56 is negative. Use the quadratic formula.

What is the vertex of y = -3x² - 8x - 10?

(-4/3, -14/3). The parabola opens downwards, so this is the maximum.

What are the sum and product of the roots?

Sum = −b/a = -8/3, product = c/a = 10/3.

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