Math ↠ Quadratic equations ↠ Mixed quadratic equations

Mixed quadratic equations

A mixed quadratic equation is an equation in the form:

$x^2+px+q=0$

Mixed quadratic equations can be solved with the help of completing the square.

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Method

  1. If a coefficient is before the $x^2$, it must be excluded beforehand, for example:
    $2x^2+4x=8$
    $2(x^2+2x)=8$

  2. Completing the square on both sides of the equation
    $x^2+px\color{red}{+(\frac{p}{2})^2}=-q+\color{red}{(\frac{p}{2})^2}$

  3. Apply binomial formula backwards $(x+\color{red}{\frac{p}{2}})^2=-q+\color{red}{(\frac{p}{2})^2}$

  4. Calculate square root and put $x$ on one side

Example

  1. Determine $p$:


    $x^2+20x=-19$
    $p=20$

  2. Completing the square $+(\frac{p}{2})^2$:


    $x^2+20x=-19\quad|+\color{red}{10^2}$
    $x^2+20x+\color{red}{10^2}=-19+\color{red}{10^2}$

  3. Apply binomial formula backwards and calculate square root


    $(x+10)^2=81\quad|\pm\sqrt{}$

  4. Put $x$ alone on one side of the equation

    There are two solutions (one positive and one negative square root)

    $x+10=9\quad|-10$
    and
    $x+10=-9\quad|-10$

  5. $x_1=-1$ and $x_2=-19$