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:

 $\displaystyle z_1$ $=$ $\displaystyle 2 + 2 i$ $\displaystyle z_2$ $=$ $\displaystyle 2 - 2 i$

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 $\displaystyle$ $=$ $\displaystyle 2 \pm \sqrt {-4}$ $\displaystyle$ $=$ $\displaystyle 2 \pm 2 i$

$\blacksquare$