The #color(blue)"Quadratic Formula"# can derived by completing the square algebraically.
From the form:
#ax^2+bx+c#
Start with:
#ax^2+bx+c=0#
Move the constant #c# to the left-hand side:
#ax^2+bx=-c#
Divide by the coefficient of #x^2#:
#a/ax^2+b/ax=-c/a#
#x^2+b/ax=-c/a#
Add to both sides the square of half the coefficient of #x#:
#x^2+b/ax+(b/(2a))^2=-c/a+(b/(2a))^2#
Simplify left hand side:
#x^2+b/ax+b/(2a)=-c/a+b^2/(4a^2)#
#x^2+b/ax+b/(2a)=(b^2-4ac)/(4a^2)#
Arrange right hand side into the square of a binomial:
#(x^2+b/(2a))^2=(b^2-4ac)/(4a^2)#
Take square roots of both sides:
#x+b/(2a)=sqrt((b^2-4ac))/sqrt((4a^2))=+-sqrt((b^2-4ac))/(2a)#
Subtract #b/(2a)# from both sides:
#x=-b/(2a)+-sqrt((b^2-4ac))/(2a)#
Add fractions:
#color(blue)(x=(-b+-sqrt((b^2-4ac)))/(2a))#