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:

\(\displaystyle z_1\) \(=\) \(\displaystyle 2 + 2 i\)
\(\displaystyle z_2\) \(=\) \(\displaystyle 2 - 2 i\)


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