Given a term of the form
it is often useful to express it as
This is often useful when solving certain quadratic equations. In our case it lets us more easily use Laplace Transform Tables (in the table the form (s+a)2+ω02 comes up frequently).
Without loss of generality, we will only consider the case where α=1. If we divide the original equation by through by α and let b=β/α and c=γ/α, we get
and our task is to express it as
We start by setting the two terms to be equal to each other
x2 + bx + c = (x+d)2 + e
Expand the right hand side
x2 + bx + c = x2 + 2dx + d2 + e
Equating the coefficients of like powers of x we get
b = 2d,
c = d2 + e
d = b/2, and
e = c - d2
Complete the square for the expression: s2+2s+10
The original function is of the form "s2 + bs + c", so b=2, c=10, and
d = b/2 = 1
e = c - d2 = 10 - 1 = 9.
The desired expression is "(s+d)2 + e" or (s+1)2+9
Complete the square for the expression: x2+4x+29
The original function is of the form "x2 + bx + c", so b=4, c=29, and
d = b/2 = 2
e = c - d2 = 29 - 4 = 25.
The desired expression is "(x+d)2 + e" or (x+2)2+25