Math Quadratic equations Vieta's formulas

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


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