Didactic Method to Complete the Square
Given a polynomial of the form $$ax^{2}+bx+c=0$$
one can follow these steps to complete the square.
-
Divide each term by the coefficient of the $x^{2}$ term and algebraically move the constant term to the right side.
$$\frac{a}{a}x^{2}+\frac{b}{a}x=\frac{-c}{a}$$
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Add the term $\left(\begin{array}{c}
\frac{b}{2a}\end{array}\right)^{2}$ to both sides.
$$x^{2}+\frac{b}{a}x+\left(\begin{array}{c}
\frac{b}{2a}\end{array}\right)^{2}=\frac{-c}{a}+\left(\begin{array}{c}
\frac{b}{2a}\end{array}\right)^{2}$$
-
Replace the left side with $(x+\frac{b}{2a})^{2}$ and combine terms on the right.
$$(x+\frac{b}{2a})^{2}=-\frac{c}{a}+\left(\begin{array}{c} \frac{b}{2a}\end{array}\right)^{2}$$
-
Solve for $x$ and get the quadratic formula.
$$x+\frac{b}{2a}=\pm\sqrt{\frac{-c}{a}+\left(\begin{array}{c}
\frac{b}{2a}\end{array}\right)^{2}}$$
$$x=\frac{-b}{2a}\pm\sqrt{\frac{-c}{a}+\left(\begin{array}{c}
\frac{b}{2a}\end{array}\right)^{2}}$$
$$x=\frac{-b}{2a}\pm\sqrt{\frac{-c}{a}\cdot\frac{4a}{4a}+\frac{b^{2}}{4a^{2}}}$$
$$x=\frac{-b}{2a}\pm\sqrt{\frac{b^{2}}{4a^{2}}-\frac{4ac}{4a^{2}}}$$
$$x=\frac{-b\pm\sqrt{b^{2}-4ac}}{2a}$$