Question 991·Hard·Math

Type $m=0$, then graph both given equations. Desmos shows three intersections. An intersection is a point on both graphs, so each one is a solution. Starting at $0$ lets you see why getting exactly two distinct intersections requires a special value of $m$; it isn't a guess for the answer.

At a shared point, both equations give the same $y$, so set their right sides equal:

Subtract $mx+2$ from both sides and combine the $x$ terms:

A root is an $x$-value that makes this equation $0$. A cubic with exactly two distinct real roots has one root that occurs twice. Call that repeated root $r$ and the other root $s$. Then the cubic must have these factors:

A coefficient is the number multiplying a power of $x$. Expand the factors to expose the $x^2$ coefficient:

The original cubic has no $x^2$ term, so set that coefficient to $0$:

Solve for $s$:

The constant term in the expansion is $-r^2s$. Replace $s$ with $-2r$ and match the original constant, $-2$:

Multiply:

Divide by $2$:

Take the real cube root:

Match the $x$ coefficients in the two forms of the cubic:

Replace $s$ with $-2r$:

Combine like terms:

Put in $r=-1$:

Multiply by $-1$:

Subtract $4$:

Here $s=-2r=2$, which differs from $r=-1$, so the system has exactly two distinct real solutions. Grid in -1.