Co-ordinate geometry (tick box quiz question) [if y=|x² +9x|+d has three solutions, d?]

Question:
image

My approach for solution: Two possible equations-

  1. y = x² + 9x + d
  2. y = -x² -9x + d

For these both i found the highest or lowest point, using: c-(b²/4a)

  1. d-(81/4)= d-20.25
  2. d-(81/4)= d-20.25

Then, i used the discriminant to find possible outcomes: b²-4ac

  1. 81-4(1)(20.25)
  2. 81-4(-1)(20.25)

Problem Statement:
While I used desmos to just understand why d=20.25 wouldn’t work. However, i still did not completely get it. (btw, the answer is d=-20.25). How would I solve this during the D-day (test day).

Thanks in advance!

Are you trying to do it graphically or algebraically?

algebraically

For |x^2 + 9x| = -d to have real solutions, the right side must be non-negative. This would force d \leq 0.

Solving the absolute value equation yields the following equations:

  1. x^2 + 9x = d

  2. x^2 + 9x = -d

To have exactly three unique solutions for x, one of these quadratic equations must have two distinct real solutions while the other has exactly one real solution (a repeated root). You would be analyzing the discriminant for the above task, which I’ll let you attempt.