"Completing the square" is how the quadratic equation is derived. If the quadratic ax^2+bx+c=0, for unknown values of a,b, and c, you arrive at:
x=(-b±√(b^2-4ac))/(2a)
You proceed in this manner:
x^2+6x+8=0 Since the leading coefficient is already one, no division is necessary. You then move the constant to the other side of the equation by subtracting 8 from both sides...
x^2+6x=-8 Then you halve the linear coefficient, square it, and add that value to both sides. (6/2)^2=9 so
x^2+6x+9=1 Now the left side is a "perfect square" equal to:
(x+3)^2=1 Now take the square root of both sides
x+3=±√1 Subtract 3 from both sides
x=-3±√1 So x is equal to:
x=-3-1 and -3+1
x=-4 and -2 or in factored form
(x+4)(x+2)
The same process for unknown a,b, and c:
ax^2+bx+c=0
x^2+bx/a+c/a=0
x^2+bx/a=-c/a
x^2+bx/a+b^2/(4a^2)=b^2/(4a^2)-c/a
x^2+bx/a+b^2/(4a^2)=(b^2-4ac)/(4a^2)
(x+b/(2a))^2=(b^2-4ac)/(4a^2)
x+b/(2a)=±√(b^2-4ac)/(4a^2)
x+b/(2a)=±√(b^2-4ac)/(2a)
x=(-b±√(b^2-4ac))/(2a)
