TMUA Master

For a quadratic equation, the discriminant gives a familiar way to determine how many real roots there are.

But what if the equation is cubic? What if it contains a modulus, a logarithm, or an exponential? What if the variable is restricted to a particular interval?

The quadratic discriminant is no longer enough. We need a way of thinking that is not tied to one particular type of equation.

Instead of asking which formula applies, ask what the function itself tells us about its solutions.

The key shift is to think about graphs and intersections.

From Roots to Intersections

Consider $x^2-2x-k=0$.

Rearranging and completing the square gives $(x-1)^2-1=k$.

Define $f(x)=(x-1)^2-1$.

The equation is now $f(x)=k$.

Instead of immediately solving for $x$, ask:

How many times does the horizontal line $y=k$ intersect the graph of $y=f(x)$?

Each intersection gives one real solution.

The graph has a minimum at $(1,-1)$, so $-1$ is the critical value at which the number of intersections can change.

So:

We could obtain the same result using the discriminant. But the graph reveals the more general idea:

The number of solutions changes when $y=k$ passes through a critical value of the function.

For this parabola, that critical value comes from its minimum.

From One Turning Point to Two

Now consider a cubic: $f(x)=x^3-3x$.

The same idea still works, even though the quadratic discriminant does not.

Differentiate: $f'(x)=3x^2-3=3(x-1)(x+1)$.

The stationary points occur at $x=-1 \quad\text{and}\quad x=1$.

Their function values are $f(-1)=2,\qquad f(1)=-2$.

So the graph has a local maximum at $(-1,2)$ and a local minimum at $(1,-2)$.

Now consider $x^3-3x=k$.

Again, keep the graph of $y=f(x)$ fixed and imagine moving the horizontal line $y=k$.

This time there are two critical values: $k=-2\quad\text{and}\quad k=2$.

The two turning-point values divide the possible values of $k$ into five cases:

The important new possibility is the middle region: $-2<k<2$.

Here the horizontal line passes through all three parts of the cubic, giving three distinct real solutions.

At $k=\pm2$, two nearby intersections have merged at a turning point. The equation still has a repeated root there, but only two distinct real solutions.

The General Pattern

The quadratic and cubic look different, but the reasoning is the same.

This is why the graphical viewpoint is more powerful than the discriminant alone. It is not tied to quadratics.

But turning points are not the only source of critical values. A domain endpoint, a modulus vertex, a maximum or minimum created by a restriction, or another structural feature of a graph may also change the number of possible intersections.

So for an equation of the form

$$f(x)=k,$$

a useful approach is:

The aim is not to avoid algebra. Algebra may help us find turning points or critical values. The important change is knowing what information we actually need.

Key Idea: Do not begin by trying to solve $f(x)=k$. First identify the critical values of the function where the number of intersections with $y=k$ can change. The same idea works far beyond quadratics.