In order to complete the square of a quadratic equation, first let's identify the notable product below: the square of a sum:
[tex](a+b)^2=a^2+2ab+b^2[/tex]So, in order to complete the square of a quadratic expression ax² + bx + c, when a = 1, we can add and subtract the term (b/2)², this way we have:
[tex]\begin{gathered} x^2+bx+c=0\\ \\ x^2+bx+(\frac{b}{2})^2-(\frac{b}{2})^2+c=0\\ \\ (x+\frac{b}{2})^2-(\frac{b}{2})^2+c=0\\ \\ (x+\frac{b}{2})^2=(\frac{b}{2})^2-c\\ \\ x+\frac{b}{2}=\pm\sqrt{(\frac{b}{2})^2-c}\\ \\ x=-\frac{b}{2}\pm\sqrt{(\frac{b}{2})^2-c}\\ \\ x=\frac{-b\pm\sqrt{b^2-4c}\\}{2} \end{gathered}[/tex]If a is not equal to 1, we can write the following general solution:
[tex]x=\frac{-b\pm\sqrt{b^2-4ac}}{2a}[/tex]