Question 15696
Given that, for all values of x: {{{f(x)=3x^2+12x+5=p(x+q)^2+r}}}

Find the values of p,q and r
  the given equation is a second degree polynomial in x and should have 2 roots or it should be satisfied by 2 values of x . but it is given that it is true for all values of x . so it is an identity . so coefficients of corresponding terms on either side of the identity shall be equal
  l.h.s.={{{f(x)=3x^2+12x+5}}}=r.h.s.={{{p(x+q)^2+r}}}={{{px^2+2*p*q*x+p*q^2+r}}}
hence 3=p
     12 = 2p*q   or...... q=12/2*p = 12/2*3 = 12/6 =2
     5= {{{p*q^2+r}}}= 3*2^2+r=12+r...or.....r=5-12 = -7