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.

i

### 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 $x_2=-19$