Goal: Transform x^2 + a*x + b to (x + c)^2 + d.
Setting the two sides equal to each other:
x^2 + a*x + b = (x + c)^2 + d
x^2 + a*x + b = x^2 + 2*c*x + c^2 + d
Next I will use a technique that calculus students normally use when decomposing partial fractions.
Setting the coefficients of x^2, x, and the constant equal to each other, we have:
x^2: 1 = 1
x: a = 2*c
constant: b = c^2 + d
Solving for c:
a = 2*c
c = a/2
Then solving for d:
b = c^2 + d
b = a^2/4 + d
d = b - a^2/4
Hence:
x^2 + a*x + b = (x + a/2)^2 + (b - a^2/4)
Hope this helps. Until next time!
Eddie
This blog is property of Edward Shore. 2013