the distinct-real-roots case
Of the three outcomes of the characteristic equation, the friendliest is when its two roots are different real numbers. This is the situation of an overdamped system — a screen door with a strong closer, a heavily-damped instrument needle — where motion just dies away without overshoot, as a blend of two pure exponential decays (or growths).
Concretely: for a y'' + b y' + c y = 0, the characteristic quadratic a r^2 + b r + c = 0 has two distinct real roots r1 and r2 whenever its discriminant b^2 - 4ac is positive. Each root contributes an exponential solution, e^(r1 x) and e^(r2 x). These two are automatically linearly independent — their ratio e^((r1 - r2)x) is non-constant because r1 ≠ r2, and their Wronskian (r2 - r1) e^((r1+r2)x) never vanishes — so they form a fundamental set. The general solution is simply y = c1 e^(r1 x) + c2 e^(r2 x), and the constants c1, c2 are fixed by two initial conditions.
Reading the roots tells you the behaviour at a glance. If both roots are negative, every solution decays to zero (a stable, overdamped return to rest). If one is positive, generic solutions blow up. If the roots straddle zero, you get a saddle-like split. This case is the simplest of the three and the template for the others, which are really limiting or complex variants of the same exponential idea.
y'' - y' - 6 y = 0 has characteristic equation r^2 - r - 6 = (r - 3)(r + 2) = 0, so r = 3, -2. The general solution is y = c1 e^(3x) + c2 e^(-2x). With one growing and one decaying mode, generic solutions grow like e^(3x) as x increases.
Two distinct real roots give two pure exponentials; their sum is the general solution.
Distinct real roots require a strictly positive discriminant. When b^2 - 4ac is exactly zero the two roots merge into one (the repeated-root case), and the second solution is not another exponential but x e^(r x).