Viète's Formulas/Examples/Sum 4, Product 8

Example of Use of Viète's Formulas

Let $z_1$ and $z_2$ be two numbers whose sum is $4$ and whose product is $8$.

Then:

\(\ds z_1\)

\(=\)

\(\ds 2 + 2 i\)

\(\ds z_2\)

\(=\)

\(\ds 2 - 2 i\)

Proof

Let $z_1$ and $z_2$ be the roots of the quadratic equation:

$z^2 + b z + c = 0$

From Viète's Formulas:

\(\ds b\)

\(=\)

\(\ds -4\)

\(\ds c\)

\(=\)

\(\ds 8\)

and so $z_1$ and $z_2$ are the roots of the quadratic equation:

\(\ds z^2 - 4 z + 8\)

\(=\)

\(\ds 0\)

\(\ds \leadsto \ \ \)

\(\ds z\)

\(=\)

\(\ds \dfrac {4 \pm \sqrt {4^2 - 4 \times 8} } 2\)

Quadratic Formula

\(\ds \)

\(=\)

\(\ds 2 \pm \sqrt {-4}\)

\(\ds \)

\(=\)

\(\ds 2 \pm 2 i\)

$\blacksquare$

Sources

• 1981: Murray R. Spiegel: Theory and Problems of Complex Variables (SI ed.) ... (previous) ... (next): $1$: Complex Numbers: Supplementary Problems: Polynomial Equations: $104$