Inverse Laplace transform with minus $\Delta$ in denominator

Question

Please help me find this inverse Laplace transform. $$ F(s)=\dfrac{2s-3}{s^{2}-2s+2} $$ I couldn't resolve the denominator, because the quadratic has discriminant $\Delta=-4$.

Answer 1

Indeed, the discriminant is negative, so there are no roots of the denominator. This means that instead of factoring it, you should complete the square: $$s^2-2s+2 = (s-1)^2+1$$ Then split the fraction as $$\frac{2(s-1)}{(s-1)^2+1}+ \frac{1}{(s-1)^2+1}$$ Consult the table of Laplace transforms and don't forget the frequency shift property.