By means of forming and application completing the square, the general solution of the quadratic equation in the form of p, q-formula are given:

${x}^{2}+px+q$$=0$

${x}^{2}+px$$=-\mathrm{q}$

${x}^{2}+px+{\left(\frac{p}{2}\right)}^{2}-{\left(\frac{p}{2}\right)}^{2}$$=-\mathrm{q}$

${x}^{2}+px+{\left(\frac{p}{2}\right)}^{2}$$={\left(\frac{p}{2}\right)}^{2}-\mathrm{q}$

${\left(x+\frac{p}{2}\right)}^{2}$$={\left(\frac{p}{2}\right)}^{2}-\mathrm{q}$

$x+\frac{p}{2}$$=\pm \sqrt{{\left(\frac{p}{2}\right)}^{2}-\mathrm{q}}$

${x}_{1,2}$$=-\frac{p}{2}\pm \sqrt{{\left(\frac{p}{2}\right)}^{2}-\mathrm{q}}$

Starting from the normal form of the quadratic equation, the equation is solved using quadratic supplement.

Starting point for the general solution is the normal form of the quadratic equation.

1. Subtraction q

2. Enhancements to the equation ${\left(\frac{p}{2}\right)}^{2}$ and subtraction of this term, so that the equation is not actually changed.

3. After the conversion is on the left side of the equation a term that corresponds to the first binomial theorem: ${\left(a+b\right)}^{2}={a}^{2}+2ab+{b}^{2}$

4. Application of the binomial leads to a quadratic expression.

5. Pull the root then allows the resolution of the equation for x. Because the square root in general has a positive and a negative quadratic equation, the solution also in general two solutions x_{1} und x_{2}.

6. Result is the so-called pq-formula for determining the solution of a quadratic equation.