Initially when I saw the graph

for x^4 + 7x^3 + 9x^2 - 7x, I noticed that if k were somewhere around -2, the lowest three roots would be in arithmetic progression, with the second root being midway between the first and the third; and that if k were somewhere around -7, the highest three roots would be in arithmetic progression.

Only later did I notice that near (or maybe exactly at) k = -10, the first, second and fourth of the roots would be in arithmetic progression, maybe at exactly -5, -2 and 1. A check with a calculator shows that indeed, k = -10 works exactly, with those exact roots.

But that still leaves values of k near -2 and -7 as valid answers as well.

(From inspection of the graph, roots 1, 3 and 4 would never be in arithmetic progression.)

        k                                  roots
-----------------  -------------------------------------------------------
-2.25893813435928  -4.88240472602988  -2.56872930440884  -.25505388278781
-6.94809311564072  -2.26932171598903  -.681270695591157   .906780324806719

nk in the below program is the negative of k, which is the height of the graph shown by the link above for the function without k.

DefDbl A-Z

Dim crlf$

Private Sub Form_Load()

 Form1.Visible = True

 

 Text1.Text = ""

 crlf = Chr$(13) + Chr$(10)

 

 low = 1.5: high = 2.5

 Do

   md = (low + high) / 2

   a = root(md, -5, -4.5)

   b = root(md, -3, -2)

   c = root(md, -0.5, 0)

   If c - b < b - a Then high = md

   If c - b > b - a Then low = md

 Loop Until high = low Or c - b = b - a

 Text1.Text = Text1.Text & -md & "  " & Str(a) & "  " & Str(b) & "  " & Str(c) & crlf

 low = 6: high = 8

 Do

   md = (low + high) / 2

   a = root(md, -3, -2)

   b = root(md, -1, 0)

   c = root(md, 0.5, 1.5)

   If c - b > b - a Then high = md

   If c - b < b - a Then low = md

 Loop Until high = low Or c - b = b - a

 Text1.Text = Text1.Text & -md & "  " & Str(a) & "  " & Str(b) & "  " & Str(c) & crlf

 Text1.Text = Text1.Text & crlf & " done"

  

End Sub

Function root(nk, btm, top)

  low = btm: high = top

  drct = Sgn(f(high, nk) - f(low, nk))

 Do

  DoEvents

   md = (low + high) / 2

   x = md

   y = f(x, nk)

   If y * drct > 0 Then high = md

   If y * drct < 0 Then low = md

 Loop Until Abs(high - low) < 0.000000000000001 Or Abs(y) < 0.000000000000001

 root = x

End Function

Function f(x, nk)

 f = x ^ 4 + 7 * x * x * x + 9 * x * x - 7 * x - nk

End Function