Solve x² + 2 = 0

x² + 2 = 0 has no real solutions. The two complex solutions are x = √2 i and x = -√2 i.

x² +x += 0

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

Discriminant D = -8

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

x^2 + 2 = 0\ \Rightarrow\ x = \sqrt{2}\, i,\ -\sqrt{2}\, i

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Graph of y = x² + 2. The curve never reaches the x-axis, which is why there are no real solutions; the vertex is at (0, 2).

How to work it out

1
Write it in standard form a x² + b x + c = 0 and read off the coefficients: a = 1, b = 0, c = 2.
$x^2 + 2 = 0$
2
Work out the discriminant D = b² − 4ac. Its sign tells you how many real solutions there are.
$D = 0^2 - 4\cdot1\cdot2 = 0 - 8 = -8$
3
D is negative, so there are no real solutions; the two solutions are complex numbers.
4
The quadratic formula works for every quadratic:
$x = \frac{-b \pm \sqrt{D}}{2a} = \frac{0 \pm \sqrt{-8}}{2}$
5
Simplify (pull the square out of the root) to get the solutions:
$x = \sqrt{2}\, i\quad\text{or}\quad x = -\sqrt{2}\, i$

LaTeX source of the formulas on this page

\[ x^2 + 2 = 0\ \Rightarrow\ x = \sqrt{2}\, i,\ -\sqrt{2}\, i \]
\[ x^2 + 2 = 0 \]
\[ D = 0^2 - 4\cdot1\cdot2 = 0 - 8 = -8 \]
\[ x = \frac{-b \pm \sqrt{D}}{2a} = \frac{0 \pm \sqrt{-8}}{2} \]
\[ x = \sqrt{2}\, i\quad\text{or}\quad x = -\sqrt{2}\, i \]

The parabola y = x² + 2

x	y
-3	11
-2	6
-1	3
0	2
1	3
2	6
3	11

Questions

What are the solutions of x² + 2 = 0?

x² + 2 = 0 has no real solutions. The two complex solutions are x = √2 i and x = -√2 i.

Does x² + 2 = 0 factor?

Not over the integers, because the discriminant -8 is negative. Use the quadratic formula.

What is the vertex of y = x² + 2?

(0, 2). 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 = 2.

Solve x² + 2 = 0

How to work it out

The parabola y = x² + 2

Questions

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