Quadratic Equation/Examples/x^2 + 1 = 0

Example of Quadratic Equation

The quadratic equation:

$x^2 + 1 = 0$

has no root in the set of real numbers $\R$:

$x = \pm i$

where $i = \sqrt {-1}$ is the imaginary unit.

Proof

From the Quadratic Formula:

\(\ds x^2 + 1\)

\(=\)

\(\ds 0\)

\(\ds \leadsto \ \ \)

\(\ds x\)

\(=\)

\(\ds \dfrac {-0 \pm \sqrt {0^2 - 4 \times 1 \times 1} } {2 \times 1}\)

Quadratic Formula

\(\ds \)

\(=\)

\(\ds \pm \sqrt {-1}\)

\(\ds \)

\(=\)

\(\ds \pm i\)

Definition of Imaginary Unit

$\blacksquare$

Sources

• 1960: Walter Ledermann: Complex Numbers ... (previous) ... (next): $\S 1.1$. Number Systems: $(1.1)$