+1222 For what values of $j$ does the equation $(2x + 7)(x - 4) = -31 + jx$ have exactly one real solution? Express your answer as a list of numbers, separated by commas.
+1897 First, we have to simplify the left side of the equation.
Expanding it out, we get
\(2x^2-x-28=-31+jx\)
Moving all terms to one side and combining all like terms, we achieve
\(2x^2-x-jx-28+31=0\\ 2x^2-\left(j+1\right)x+3=0\)
If there's only one solution, then that must mean that the descriminant is 0.
The descriminant of this quadratic is
\((j+1)^2 - 4(2)(3)\\ j^2+2j-23\)
Setting this to 0, we can now solve for j. We get
\(j^2+2j-23=0\)
Unfortunately, j cannot be factored conveniently, so we must use the quadratic equation. (I'm going to leave that for you).
From that, we find that
\(j=-1+2\sqrt{6},\:j=-1-2\sqrt{6}\)
I'm pretty sure this is the correct answer, but you can check and verify.
Thanks! :)
