diff --git a/doc/source/whatsnew/v0.20.0.txt b/doc/source/whatsnew/v0.20.0.txt
index ece9ff4a1adff..8dbc5673f1bcd 100644
--- a/doc/source/whatsnew/v0.20.0.txt
+++ b/doc/source/whatsnew/v0.20.0.txt
@@ -670,8 +670,8 @@ Bug Fixes
 - Bug in ``DataFrame.loc`` with indexing a ``MultiIndex`` with a ``Series`` indexer (:issue:`14730`, :issue:`15424`)
 - Bug in ``DataFrame.loc`` with indexing a ``MultiIndex`` with a numpy array (:issue:`15434`)
 - Bug in ``Rolling.quantile`` function that caused a segmentation fault when called with a quantile value outside of the range [0, 1] (:issue:`15463`)
-
-
+- Bug in ``pd.cut()`` single bin on all 0s array raises ``ValueError`` (:issue:`15428`)
+- Bug in ``pd.qcut()`` single quantile and array with identical values raises ``ValueError`` (:issue:`15431`)
 - Bug in ``SparseSeries.reindex`` on single level with list of length 1 (:issue:`15447`)
 
 
diff --git a/pandas/tests/tools/test_tile.py b/pandas/tests/tools/test_tile.py
index de44eadc15751..11b242bc06e15 100644
--- a/pandas/tests/tools/test_tile.py
+++ b/pandas/tests/tools/test_tile.py
@@ -3,7 +3,7 @@
 import numpy as np
 from pandas.compat import zip
 
-from pandas import Series, Index
+from pandas import Series, Index, Categorical
 import pandas.util.testing as tm
 from pandas.util.testing import assertRaisesRegexp
 import pandas.core.common as com
@@ -239,7 +239,6 @@ def test_qcut_binning_issues(self):
             self.assertTrue(ep <= sn)
 
     def test_cut_return_categorical(self):
-        from pandas import Categorical
         s = Series([0, 1, 2, 3, 4, 5, 6, 7, 8])
         res = cut(s, 3)
         exp = Series(Categorical.from_codes([0, 0, 0, 1, 1, 1, 2, 2, 2],
@@ -249,7 +248,6 @@ def test_cut_return_categorical(self):
         tm.assert_series_equal(res, exp)
 
     def test_qcut_return_categorical(self):
-        from pandas import Categorical
         s = Series([0, 1, 2, 3, 4, 5, 6, 7, 8])
         res = qcut(s, [0, 0.333, 0.666, 1])
         exp = Series(Categorical.from_codes([0, 0, 0, 1, 1, 1, 2, 2, 2],
@@ -285,6 +283,60 @@ def test_qcut_duplicates_bin(self):
         # invalid
         self.assertRaises(ValueError, qcut, values, 3, duplicates='foo')
 
+    def test_single_quantile(self):
+        # issue 15431
+        expected = Series([0, 0])
+
+        s = Series([9., 9.])
+        result = qcut(s, 1, labels=False)
+        tm.assert_series_equal(result, expected)
+        result = qcut(s, 1)
+        exp_lab = Series(Categorical.from_codes([0, 0], ["[9, 9]"],
+                                                ordered=True))
+        tm.assert_series_equal(result, exp_lab)
+
+        s = Series([-9., -9.])
+        result = qcut(s, 1, labels=False)
+        tm.assert_series_equal(result, expected)
+        result = qcut(s, 1)
+        exp_lab = Series(Categorical.from_codes([0, 0], ["[-9, -9]"],
+                                                ordered=True))
+        tm.assert_series_equal(result, exp_lab)
+
+        s = Series([0., 0.])
+        result = qcut(s, 1, labels=False)
+        tm.assert_series_equal(result, expected)
+        result = qcut(s, 1)
+        exp_lab = Series(Categorical.from_codes([0, 0], ["[0, 0]"],
+                                                ordered=True))
+        tm.assert_series_equal(result, exp_lab)
+
+        expected = Series([0])
+
+        s = Series([9])
+        result = qcut(s, 1, labels=False)
+        tm.assert_series_equal(result, expected)
+        result = qcut(s, 1)
+        exp_lab = Series(Categorical.from_codes([0], ["[9, 9]"],
+                                                ordered=True))
+        tm.assert_series_equal(result, exp_lab)
+
+        s = Series([-9])
+        result = qcut(s, 1, labels=False)
+        tm.assert_series_equal(result, expected)
+        result = qcut(s, 1)
+        exp_lab = Series(Categorical.from_codes([0], ["[-9, -9]"],
+                                                ordered=True))
+        tm.assert_series_equal(result, exp_lab)
+
+        s = Series([0])
+        result = qcut(s, 1, labels=False)
+        tm.assert_series_equal(result, expected)
+        result = qcut(s, 1)
+        exp_lab = Series(Categorical.from_codes([0], ["[0, 0]"],
+                                                ordered=True))
+        tm.assert_series_equal(result, exp_lab)
+
     def test_single_bin(self):
         # issue 14652
         expected = Series([0, 0])
@@ -297,6 +349,29 @@ def test_single_bin(self):
         result = cut(s, 1, labels=False)
         tm.assert_series_equal(result, expected)
 
+        expected = Series([0])
+
+        s = Series([9])
+        result = cut(s, 1, labels=False)
+        tm.assert_series_equal(result, expected)
+
+        s = Series([-9])
+        result = cut(s, 1, labels=False)
+        tm.assert_series_equal(result, expected)
+
+        # issue 15428
+        expected = Series([0, 0])
+
+        s = Series([0., 0.])
+        result = cut(s, 1, labels=False)
+        tm.assert_series_equal(result, expected)
+
+        expected = Series([0])
+
+        s = Series([0])
+        result = cut(s, 1, labels=False)
+        tm.assert_series_equal(result, expected)
+
     def test_datetime_cut(self):
         # GH 14714
         # testing for time data to be present as series
diff --git a/pandas/tools/tile.py b/pandas/tools/tile.py
index feb4d4bfd5044..034ecd72bd41e 100644
--- a/pandas/tools/tile.py
+++ b/pandas/tools/tile.py
@@ -104,8 +104,8 @@ def cut(x, bins, right=True, labels=None, retbins=False, precision=3,
         mn, mx = [mi + 0.0 for mi in rng]
 
         if mn == mx:  # adjust end points before binning
-            mn -= .001 * abs(mn)
-            mx += .001 * abs(mx)
+            mn -= .001 * abs(mn) if mn != 0 else .001
+            mx += .001 * abs(mx) if mx != 0 else .001
             bins = np.linspace(mn, mx, bins + 1, endpoint=True)
         else:  # adjust end points after binning
             bins = np.linspace(mn, mx, bins + 1, endpoint=True)
@@ -206,7 +206,7 @@ def _bins_to_cuts(x, bins, right=True, labels=None,
                          "valid options are: raise, drop")
 
     unique_bins = algos.unique(bins)
-    if len(unique_bins) < len(bins):
+    if len(unique_bins) < len(bins) and len(bins) != 2:
         if duplicates == 'raise':
             raise ValueError("Bin edges must be unique: {}.\nYou "
                              "can drop duplicate edges by setting "