Spotting where a root hides

Some equations are impossible to solve exactly. For example, there is no neat formula for the root of

$x^3 + x - 3 = 0.$

Cambridge does not ask you to invent one. It asks this instead:

Show that the equation $x^3 + x - 3 = 0$ has a root between $1$ and $2$.

"A root between 1 and 2" means: somewhere in that interval, the curve crosses the $x$-axis. You do not need to say where. You only need to prove it must happen.

Write the equation as $f(x) = 0$, then follow four steps.

1. Name the function. Move everything to one side: $f(x) = x^3 + x - 3$.
2. Evaluate at both ends. Work out $f(1)$ and $f(2)$. Don't round so far that a small value turns into 0: the sign is what matters.
3. Compare the signs. One value is negative and the other is positive.
4. Write the conclusion. Say "$f(1) < 0$ and $f(2) > 0$, and $f$ is continuous, so there is a root between $1$ and $2$."

Why it works: a continuous curve cannot get from below the $x$-axis to above it without passing through it. Changing sign forces a crossing.

The forgotten step: that word continuous. A curve with a break can jump from negative to positive without ever being zero, so the conclusion needs continuity stated (or a graph that plainly has no break).