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

< Viète's Formulas | Examples

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## 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:

\(\displaystyle z_1\) | \(=\) | \(\displaystyle 2 + 2 i\) | |||||||||||

\(\displaystyle z_2\) | \(=\) | \(\displaystyle 2 - 2 i\) |

## Proof

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

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

From Viète's Formulas:

\(\displaystyle b\) | \(=\) | \(\displaystyle -4\) | |||||||||||

\(\displaystyle c\) | \(=\) | \(\displaystyle 8\) |

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

\(\displaystyle z^2 - 4 z + 8\) | \(=\) | \(\displaystyle 0\) | |||||||||||

\(\displaystyle \leadsto \ \ \) | \(\displaystyle z\) | \(=\) | \(\displaystyle \dfrac {4 \pm \sqrt {4^2 - 4 \times 8} } 2\) | Quadratic Formula | |||||||||

\(\displaystyle \) | \(=\) | \(\displaystyle 2 \pm \sqrt {-4}\) | |||||||||||

\(\displaystyle \) | \(=\) | \(\displaystyle 2 \pm 2 i\) |

$\blacksquare$

## Sources

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