Question

Multiply the equation by 4p. Explain how adding (x – h)2 to the 3 values of k verifies the number of zeros of the polynomial. Assume p>0.

Answer 1

1/(4p) (x−h)^2+k = 0 ... step 1 multiply by 4p 
(x−h)^2 + 4pk = 0 (assuming p is positive) 
(x−h)^2 = -4pk 
x - h = +/- sq rt (-4pk) 
case 1 ... k is pos. then -4pk is neg ... sq rt gives 2 complex roots 

case 2 ... k is neg. then -4pk is positive ... sq rt gives 2 real roots 

case 3 ... k = 0 then -4pk = 0 ... sq rt = 0 ... 1 real root .. x = h 

Actually, since p can be neg. the directions should say --- 3 values of (-4pk) verifies the number of zeros of the polynomial.