an algebra question

hello.

i have got this one algebra question for homework.

however i had no idea on how to do this attached one.

i need to find answers for both a= and b=.

One way is to expand the righthand side, then match its terms to those of the lefthand side.

x^2 +6x -3 = (x+a)^2 +b
x^2 +6x -3 = (x^2 +2ax +a^2) +b
x^2 +6x -3 = x^2 +2ax +a^2 +b
x^2 +6x -3 = x^2 +2ax +(a^2 +b)

So,
x^2 = x^2 ----(i)
6x = 2ax -----(ii)
-3 = (a^2 +b) -----(iii)

From (ii),
6 = 2a
a = 6/2 = 3 ----***

Substitute that into (iii),
-3 = (3)^2 +b
-3 = 9 +b
-3 -9 = b
b = -12 ---***

Check those in the original equation,
x^2 +6x -3 = (x+a)^2 +b
x^2 +6x -3 =? (x+3)^2 +(-12)
x^2 +6x -3 =? (x^2 +6x +9) -12
x^2 +6x -3 =? x^2 +6x +9 -12
x^2 +6x -3 =? x^2 +6x -3
Yes, so, OK

Therefore, a = 3 and b = -12. -------answer.

thank you.

u have given me an excellent description. thanks. :)