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What are the zeros of the function f(x)=25x^2-16

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The trick here is to find the factors of the coefficients. For 25 it is (5 x 5) or (-5 x - 5). The factors for -16 are -4 and 4 or (-2 X 8) or (2 x -8). 
Now because there is no term with just an x, that is a hint that when this equation was FOILed out the x terms were equal but in opposite directions. Looking at the factors for -16 we can see two such values in 4 and -4 (they are equal and in opposite direction). 
Based on this reasoning, one of the two solutions is f(x) = (5x -4)(5x + 4) [incidentally you could use (-5x - 4)(-5x +4) and you will get the same answer] 
FOIL it back out to double check that you get 25x- 16 
Set each value within the parentheses to 0 and then solve for x. 
(5x - 4) = 0 
(5x + 4) = 0 
I hope this helps.