Quadratic Equation/Examples/5z^2 + 2z + 10 = 0

The quadratic equation in $\C$:

$5 z^2 + 2 z + 10 = 0$

has the solutions:

$z = \dfrac {-1 \pm 7 i} 5$

Proof

\(\ds 5 z^2 + 2 z + 10\)

\(=\)

\(\ds 0\)

\(\ds \leadsto \ \ \)

\(\ds z\)

\(=\)

\(\ds \dfrac {-2 \pm \sqrt {2^2 - 4 \times 5 \times 10} } {2 \times 5}\)

Quadratic Formula: $a = 5$, $b = 2$, $c = 10$

\(\ds \)

\(=\)

\(\ds \dfrac {-1 \pm \sqrt {1 - 50} } 5\)

dividing top and bottom by $2$

\(\ds \)

\(=\)

\(\ds \dfrac {-1 \pm 7 i} 5\)

$\blacksquare$

Sources

• 1981: Murray R. Spiegel: Theory and Problems of Complex Variables (SI ed.) ... (previous) ... (next): $1$: Complex Numbers: Supplementary Problems: Polynomial Equations: $100 \ \text{(a)}$