The Ultimate Crash Course for STEM Majors
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Complete the Square
$$x^2-\frac{1}{2}x-\frac{3}{16}=0,\qquad 16x^2-8x-3=0$$
Complete the Square
$$ax^2+b+c=0$$
$$a\left(x^2+\frac{b}{a}x+\frac{c}{a}\right)=0
\Rightarrow
a\left(x^2+\frac{b}{a}x+\frac{c}{a}+0\right)=0$$
$$a\left(x^2+\frac{b}{a}x+\frac{c}{a}
+\left[\left(\frac{b}{2a}\right)^2-\left(\frac{b}{2a}\right)^2\right]\right)=0$$
$$a\left(x^2+\frac{b}{a}x+\frac{c}{a}
+\left(\frac{b}{2a}\right)^2-\left(\frac{b}{2a}\right)^2\right)=0$$
$$a\left(x^2+\frac{b}{a}x+\left(\frac{b}{2a}\right)^2
+\frac{c}{a}-\left(\frac{b}{2a}\right)^2\right)=0$$
$$a\left(x^2+\frac{b}{a}x+\left(\frac{b}{2a}\right)^2
+\frac{c}{a}-\left(\frac{b}{2a}\right)^2\right)=0$$
NOTE* Perfect square: $$(x+y)^2$$
Not perfect square: $$x^2+y^2$$
Factor: $$x^2-y^2=(x-1)(x+y)$$
$$
a\left(
{\color{red}{x^2+\frac{b}{a}x+\left(\frac{b}{2a}\right)^2}}
+\frac{c}{a}
-\left(\frac{b}{2a}\right)^2
\right)=0
$$
$$
a\left(
{\color{red}{xx}}
+
{\color{green}{\frac{1}{2}\frac{a}{b}x}}
+
{\color{green}{\frac{1}{2}\frac{a}{b}x}}
+
{\color{red}{\left(\frac{b}{2a}\right)\left(\frac{b}{2a}\right)}}
+\frac{c}{a}
-\left(\frac{b}{2a}\right)^2
\right)=0
$$
$$
a\left(
{\color{magenta}{\left[x+\frac{b}{2a}\right]^2}}
+\frac{c}{a}
-\left(\frac{b}{2a}\right)^2
\right)=0
\Rightarrow
a{\color{magenta}{\left[x+\frac{b}{2a}\right]^2}}
+
a\left[
\frac{c}{a}
-\left(\frac{b}{2a}\right)^2
\right]=0
$$
$$
\Rightarrow
a{\color{magenta}{\left[x+\frac{b}{2a}\right]^2}}
=
-a\left[
\frac{c}{a}
-\left(\frac{b}{2a}\right)^2
\right]
\Rightarrow
{\color{magenta}{\left[x+\frac{b}{2a}\right]^2}}
=
-\frac{c}{a}
+\left(\frac{b}{2a}\right)^2
=
\left(\frac{b}{2a}\right)^2-\frac{c}{a}
$$
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