alrighty
we will complete the squaer for ax²+bx+c=0 regardless of the values of a,b, or c
we will factor it into a(x-h)²=r
so
first group x terms
(ax²+bx)+c=0
factor out a
[tex]a(x^2+\frac{b}{a}x)+c=0[/tex]
take 1/2 of the linear coefinet and squaer it
[tex]\frac{1}{2} \space\ of \space\ \frac{b}{a}=\frac{b}{2a}[/tex] square it to get [tex]\frac{b^2}{4a^2}[/tex]
add positive and negative of that inside parntheasees
[tex]a(x^2+\frac{b}{a}x+\frac{b^2}{4a^2}-\frac{b^2}{4a^2})+c=0[/tex]
factor perfect square
[tex]a((x+\frac{b}{2a})^2-\frac{b^2}{4a^2})+c=0[/tex]
expand
[tex]a(x+\frac{b}{2a})^2-\frac{b^2}{4a}+c=0[/tex]
and that's how you complete the square, just move the constants over to the left when you're done then divide both sides by a then square root both sides, remembering to take the positive and negative roots
