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:
$-p=x_1+x_2$
$q=x_1\cdot x_2$
$q=x_1\cdot x_2$
Example
Using Vieta's formula, construct a quadratic equation in the canonical form that has the solutions $x_1=-1$ and $x_2=-5$.
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Determine $p$ with Vieta's formula
$-p=x_1+x_2$
$-p=-1+-5$
$-p=-6$
$p=6$ -
Determine $q$ with Vieta's formula
$q=x_1\cdot x_2$
$q=-1\cdot-5$
$q=5$ -
Insert $p$ and $q$ in the canonical form
$x^2+px+q=0$
$x^2+6x+5=0$