Find the two values of $k$ for which $2x^3-9x^2+12x-k$ has a double real root.

Here is an easier way. If your polynomial $P_k(x) = 2x^3-9x^2+12x-k$ has a double real root $r$, then $P_k'(r) = 0$.

For such a root, you get $P_k'(r) = 6r^2-18r+12 = 0$, i.e. $r=1$ or $r=2$.

Then you solve $P_k(r)=0$ with respect to $k$. More precisely :

  • If $r=1$, one needs $P_k(1)=5-k=0$ i.e. $k=5$.
  • If $r=2$, one has $P_k(2)=4-k=0$ i.e. $k=4$.

Tags:

Polynomials

Algebra Precalculus

Roots

Cubic Equations

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