Given a term of the form
αx2+βx+γ
it is often useful to express it as
α(x+δ)2+ε
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
x2+bx+c
and our task is to express it as
(x+d)2+e
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
or
d = b/2, and
e = c - d2
Complete the square for the expression: s2+2s+10
Solution:
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
Solution:
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
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