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AMC 12 focus

Equations with Parameters

Reasoning about how many solutions an equation has as a parameter varies — the discriminant, one level up.

A parameter problem asks something like “for which values of kk does this equation have two distinct real solutions?” rather than asking you to solve the equation directly. These problems almost always reduce to the same discriminant classification from the quadratics lesson — applied to an equation whose coefficients themselves involve the parameter.

The general approach

  1. Rewrite the equation in standard form, treating the parameter as a constant alongside a,b,ca, b, c.
  2. Compute the discriminant Δ=b2−4ac\Delta = b^2-4ac — it will now be an expression in the parameter.
  3. Translate the desired root condition into an inequality (or equation) on Δ\Delta, then solve that for the parameter.
Δ>0⇒2 distinct real rootsΔ=0⇒1 repeated rootΔ<0⇒0 real roots\Delta > 0 \Rightarrow \text{2 distinct real roots} \qquad \Delta = 0 \Rightarrow \text{1 repeated root} \qquad \Delta < 0 \Rightarrow \text{0 real roots}

Watch for a subtlety: if the leading coefficient aa itself contains the parameter, check separately what happens when a=0a=0 — the equation degenerates from quadratic to linear, and the discriminant analysis no longer applies to that case.

Worked example

Problem: For how many integer values of kk doeskx2−6x+k=0kx^2 - 6x + k = 0 have two distinct real roots?

Solution: First, we need k≠0k \neq 0 for this to actually be quadratic. The discriminant is:

Δ=(−6)2−4(k)(k)=36−4k2\Delta = (-6)^2 - 4(k)(k) = 36 - 4k^2

We need Δ>0\Delta > 0:

36−4k2>0  ⟹  k2<9  ⟹  −3<k<336 - 4k^2 > 0 \;\Longrightarrow\; k^2 < 9 \;\Longrightarrow\; -3 < k < 3

Combined with k≠0k \neq 0, the integer values are k∈{−2,−1,1,2}k \in \{-2,-1,1,2\} — four values.

Practice problems

1.For what value(s) of kk does x2−kx+9=0x^2 - kx + 9 = 0 have exactly one real solution?

2.For what values of mm does x2+4x+m=0x^2 + 4x + m = 0 have no real solutions?

3.For what values of cc does 2x2−4x+c=02x^2 - 4x + c = 0 have two distinct real roots?

4.For what integer values of kk does kx2−4x+1=0kx^2 - 4x + 1 = 0 have real solutions (allow one or two, and remember to check k=0k=0 separately)?

Complex Numbers