Why a completed square gives an extremum

Write ax2+bx+c=a(x-h)2+k, where a 0. The square is at least zero: a>0 gives minimum k, and a<0 gives maximum k, at x=h. The identity x2+2px=(x+p)2-p2 adds and subtracts the same number. Factor a from the quadratic and linear terms before using it. Factor the leading coefficient, halve the coefficient of x inside the brackets, complete the square, then multiply the correction by the outside coefficient. For an equation, isolate the square and take both square-root signs. Example 2x2-8x+3=2[(x-2)2-4]+3 =2(x-2)2-5 The minimum value is -5, attained at x=2. For 2x2-8x+3=0, (x-2)2= 52, so x=2+- sqrt(10)2. Exam check: the minimum value is one number; the minimum point is an ordered pair. Check that a stated domain contains the vertex before using it.