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.

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.