The system asks where a line $y = mx + 2$ intersects the cubic $y = x^3 - 4x$. For exactly two distinct solutions, the line must be tangent to the cubic at one point and cross it at another.

Graph both functions with $m$ as a slider. Adjust $m$ until the line is tangent to the cubic at one point while crossing at another.

In Desmos, enter y = x^3 - 4x to see the cubic curve.

Type y = mx + 2. Desmos will prompt you to add a slider for m.

Move the m slider and watch how the line intersects the cubic. Look for the value of m where the line just touches the cubic at one point (tangent) while crossing at another.

When you see exactly two intersection points (one tangent, one crossing), read the value of m from the slider. Click on the intersection points to confirm there are only two.

The intersection points occur where $x^3 - 4x = mx + 2$.

Rearranging: $x^3 - 4x - mx - 2 = 0$, or $x^3 - (m+4)x - 2 = 0$.

The real solutions of this cubic are the x-coordinates of the intersection points.

A cubic equation can have:

• Three distinct real roots
• One real root and two complex roots
• A double root and one simple root

For exactly two distinct real solutions, the cubic must have a double root (where the line is tangent to the cubic) and one simple root (where the line crosses).

In Desmos:

1. Graph y = x^3 - 4x
2. Graph y = mx + 2 with a slider for $m$
3. Adjust the slider until the line is tangent to the cubic at one point

You'll find that when $m = -1$, the line $y = -x + 2$ is tangent to the cubic at one point and crosses at another, giving exactly two intersection points.

With $m = -1$, the equation becomes $x^3 - 3x - 2 = 0$.

This factors as $(x+1)^2(x-2) = 0$.

The solutions are $x = -1$ (double root, tangent point) and $x = 2$ (simple root, crossing point).

So there are exactly two distinct real solutions, confirming $m = -1$.