Answer: Coreopsis is prone to powdery mildew, especially when it's planted in an area where it doesn't get full sun.
While the disease is unsightly, it doesn't cause any real harm to many of its victims (though it will damage some plants.)
Here are some general rules for helping control fungal diseases. Start by making sure that your plants are getting
enough direct sunlight.You'll also want to make sure that there's enough room between plants for air to circulate freely. Overcrowding not only makes plants more susceptible to diseases, if leaves touch other plants, those diseases can easily be spread.
In the case of powdery mildew, you can actually inhibit infection with periodic strong sprays of water (not so strong as to damage the plant.) Some people have had good luck with pesticides containing neem, a plant extract. You can also try using a homemade baking soda spray--mix 1 teaspoon baking soda per quart of water; add 1/4 tsp mild soap to help it stick to the leaves. Apply this once a week, and hose down plants between sprayings. If you choose to use a fungicide, be sure it's labeled for powdery mildew, since some are not effective against this disease.
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