Question

Consider the Laplace transform

X(s)= 12/ s2+4s+5
"Complete the square" in the denominator to rewrite X(s) in the form:
X(s)= αs​ / (s+β)2+γ

Answer 1

To rewrite the Laplace transform expression, we complete the square in the denominator.

Step-by-step explanation:

To rewrite the given Laplace transform expression in the form X(s) = αs / (s+β)²+γ, we need to complete the square in the denominator.

First, let's consider the denominator expression: s² + 4s + 5. To complete the square, we need to add and subtract (4/2)² = 4 to the expression:

s² + 4s + 4 - 4 + 5 = (s+2)² + 1

Now, the denominator is in the form (s+β)²+γ, where β = 2 and γ = 1. Therefore, the rewritten expression becomes X(s) = 12s / (s+2)²+1.