Viète's Formulas/Examples/Sum 4, Product 8
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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:
\(\ds z_1\) | \(=\) | \(\ds 2 + 2 i\) | ||||||||||||
\(\ds z_2\) | \(=\) | \(\ds 2 - 2 i\) |
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: $104$