Vieta's formulas

Vieta's formulasis helpful when two of the four variables $p$, $q$, $x_1$ and $x_2$ are given and the other two are searched.

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Remember

Vieta's formula may only be applied to quadratic equations in the canonical form (the $x^2$ in the equation is only multiplied by 1).

Given is a quadratic equation in the canonical form: $x^2+px+q=0$.
Vieta's formula says:

$-p=x_1+x_2$
$q=x_1\cdot x_2$

Example

Using Vieta's formula, construct a quadratic equation in the canonical form that has the solutions $x_1=-1$ and $x_2=-5$.

1. Determine $p$ with Vieta's formula

$-p=x_1+x_2$
$-p=-1+-5$
$-p=-6$
$p=6$
2. Determine $q$ with Vieta's formula

$q=x_1\cdot x_2$
$q=-1\cdot-5$
$q=5$
3. Insert $p$ and $q$ in the canonical form

$x^2+px+q=0$
$x^2+6x+5=0$