Math ↠ Quadratic equations ↠ Quadratic formula I

Quadratic formula I

All quadratic equations can be solved with the quadratic formula I, without having to use the elaborate completing the square, for example.

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Remember

The quadratic formula I 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: $x^2+\color{green}{p}x+\color{blue}{q}=0$.
The quadratic formula I for solving this equation is:

$x_{1,2} = -\frac{\color{green}{p}}{2} \pm\sqrt{(\frac{\color{green}{p}}{2})^2-\color{blue}{q}}$

Example

Quadratic equation in canonical form: $x^2+\color{green}{6}x+\color{blue}{5}=0$

  1. Insert $p$ and $q$ in the quadratic formula:


    $x_{1,2} = -\frac{\color{green}{p}}{2} \pm\sqrt{(\frac{\color{green}{p}}{2})^2-\color{blue}{q}}$
    $x_{1,2} = -\frac{\color{green}{6}}{2} \pm\sqrt{(\frac{\color{green}{6}}{2})^2-\color{blue}{5}}$

  2. Simplify term


    $x_{1,2} = -3 \pm\sqrt{3^2-5}$
    $x_{1,2} = -3 \pm\sqrt{4}$
    $x_{1,2} = -3 \pm2$

  3. Calculate solutions


    $x_{1} = -3+2=-1$
    $x_{2} = -3-2=-5$