+779 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.
+1804 First, let's simplfy our equation. We have
\(2x^2 - 3x - 35 = -43 + jx \\ 2x^2 - 3x - jx + 8 = 0 \\ 2x^2 - (3 + j)x + 8 = 0\)
If there is only one solution, then the descriminant of the quadratic must be 0. So, we can write the equation
\((3 + j)^2 - 4(2)(8) = 0 \\ (j + 3)^2 - 64 = 0 \\ [ (j + 3) - 8 ] [ (j + 3) + 8 ] = 0 \\ [ j - 5 ] [ j + 11] = 0\)
Thus, we have that
\(j = 5 \\ j = -11\)
So our final answer is 5 and -11.
Thanks! :)
