Natural frequencies of a bandpass filter
When in doubt, plot! Matlab code :

w_array = -10:1:10;
w0_array = -10:1:10;
wl_array_1 = zeros(length(w_array), length(w0_array));
wl_array_2 = zeros(length(w_array), length(w0_array));
syms wl

for i=1:length(w_array)
    w = w_array(i);
    for j=1:length(w0_array)
        w0 = w0_array(j);
        s1 = solve(wl^2 - w0*w*wl/2.45 - w0^2 == 0, wl, 'PrincipalValue', true);
        if isempty(s1)
            s1 = NaN;
        end
        wl_array_1(i,j) = double(s1);
        s2 = solve(w - 2.45*abs(wl/w0 - w0/wl) == 0, wl, 'PrincipalValue', true);
        if isempty(s2)
            s2 = NaN;
        end
        wl_array_2(i,j) = double(s2);
    end
end

figure(1)
for j=1:length(w0_array)
    plot(w_array,wl_array_1(:,j)), hold on
end
xlabel('\omega'), ylabel('\omega´'), xlim([-10,10]), ylim([-45,5])
title('Not considering the absolute in equation'), hold off,

figure(2)
for j=1:length(w0_array)
    plot(w_array,wl_array_2(:,j)), hold on
end
xlabel('\omega'), ylabel('\omega´'), xlim([-10,10]), ylim([-45,5])
title('Considering the absolute in equation'), hold off




As can be seen, the absolute ensures that solutions only exist for positive values of \$\omega\$ and \$\omega_0\$. Removing the absolute, solutions exist for negative \$\omega\$ and \$\omega_0\$. Now it's up to you to decide what it means and if this is important.