Answer:
(c) -2/5
Step-by-step explanation:
A "zero" or "root" or "x-intercept" of a function is a value of the independent variable (x) that makes the function (f(x)) have a value of zero. It is found where the graph of the function crosses the x-axis.
A quadratic function will have two (2) zeros. If they are different, one will be smaller than the other (leftmost on the number line).
Difference of squares
You may recognize that 25x² and 4 are both perfect squares, (5x)² and 2², respectively. That means the expression (25x² -4) is the difference of squares.
The difference of squares is a special polynomial form, useful for the way it factors:
a² -b² = (a -b)(a +b)
Here, that means the factored form of f(x) is ...
f(x) = (5x -2)(5x +2)
Zero product rule
The "zero product rule" says a product can only be zero if one or more factors is zero. This is exceptionally useful for determining the zeros of a polynomial function from its factored form.
It tells us f(x) = 0 only when one of (5x-2) = 0, or (5x+2) = 0. These are simple equations to solve to find the corresponding values of x:
5x -2 = 0 ⇒ 5x = 2 ⇒ x = 2/5
5x +2 = 0 ⇒ 5x = -2 ⇒ x = -2/5
Zeros of f(x)
The above tells us the zeros of f(x) are -2/5 and 2/5. These values of x make f(x) = 0. The smaller zero is the one that is more negative: -2/5.
smaller zero: -2/5
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Additional comment
A polynomial may have complex zeros. Only the real zeros will be x-intercepts on a graph of the function. Complex zeros must be found another way, not by graphing.
