Solve 3x² + 8 = 0

3x² + 8 = 0 has no real solutions. The two complex solutions are x = (2√6 i)/3 and x = (-2√6 i)/3.

x² +x += 0

3x^2 + 8 = 0 \Rightarrow x = \frac{2\sqrt{6}\, i}{3},\ \frac{-2\sqrt{6}\, i}{3}\ \text{(complex)}

Discriminant D = -96

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

3x^2 + 8 = 0\ \Rightarrow\ x = \frac{2\sqrt{6}\, i}{3},\ \frac{-2\sqrt{6}\, i}{3}

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Graph of y = 3x² + 8. The curve never reaches the x-axis, which is why there are no real solutions; the vertex is at (0, 8).

How to work it out

1
Write it in standard form a x² + b x + c = 0 and read off the coefficients: a = 3, b = 0, c = 8.
$3x^2 + 8 = 0$
2
Work out the discriminant D = b² − 4ac. Its sign tells you how many real solutions there are.
$D = 0^2 - 4\cdot3\cdot8 = 0 - 96 = -96$
3
D is negative, so there are no real solutions; the two solutions are complex numbers.
4
The quadratic formula works for every quadratic:
$x = \frac{-b \pm \sqrt{D}}{2a} = \frac{0 \pm \sqrt{-96}}{6}$
5
Simplify (pull the square out of the root) to get the solutions:
$x = \frac{2\sqrt{6}\, i}{3}\quad\text{or}\quad x = \frac{-2\sqrt{6}\, i}{3}$

LaTeX source of the formulas on this page

\[ 3x^2 + 8 = 0\ \Rightarrow\ x = \frac{2\sqrt{6}\, i}{3},\ \frac{-2\sqrt{6}\, i}{3} \]
\[ 3x^2 + 8 = 0 \]
\[ D = 0^2 - 4\cdot3\cdot8 = 0 - 96 = -96 \]
\[ x = \frac{-b \pm \sqrt{D}}{2a} = \frac{0 \pm \sqrt{-96}}{6} \]
\[ x = \frac{2\sqrt{6}\, i}{3}\quad\text{or}\quad x = \frac{-2\sqrt{6}\, i}{3} \]

The parabola y = 3x² + 8

x	y
-3	35
-2	20
-1	11
0	8
1	11
2	20
3	35

Questions

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

3x² + 8 = 0 has no real solutions. The two complex solutions are x = (2√6 i)/3 and x = (-2√6 i)/3.

Does 3x² + 8 = 0 factor?

Not over the integers, because the discriminant -96 is negative. Use the quadratic formula.

What is the vertex of y = 3x² + 8?

(0, 8). The parabola opens upwards, so this is the minimum.

What are the sum and product of the roots?

Sum = −b/a = 0, product = c/a = 8/3.

Solve 3x² + 8 = 0

How to work it out

The parabola y = 3x² + 8

Questions

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