When she said ax^2 + bx + c, when "a" does not equal 1, doesn't she mean when "a" does not equal 0? 

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No, if it equals 0, she wouldnt write the x. If the number before x is 0, you do not write the x, because of the zero product property, anything multiplied by zero is zero. So if the number before x is 1, 1*x=x. So you dont really need to write the 1 since it doesnt really matter.