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The zeros or roots of a function tell us where a graph intersects the x-axis. Since we’re talking about intersection with the x-axis, you know that y=0. That means that you can find the zeros by solving the equation f(x)=0.

Rule

Zeros

You find the zeros of a function by solving the equation

f(x)=0.

The sign chart of f(x) tells you when the graph of f is above or below the x-axis, and where f(x) intersects the x-axis.

Graph showing f(x) and its zeros

Example 1

Find the zeros of f(x)=x2+5x+6

You find the zeros by solving the equation f(x)=x2+5x+6=0:

x=5±254162=5±25242=5±12

This gives

x1=512=3,x2=5+12=2.

So the graph meets the x-axis at the points i (3,0) and (2,0).

Example 2

Find the zeros of g(x)=2sin(2x)+1 in the interval x[0,π)

Again, you can find the zeros by solving the equation g(x)=0:

2sin(2x)+1=0sin(2x)=12

The basic equation has the solutions

2x=π6+2πn,2x=π(π6)+2πn.

Solving these for x, you get

x=π12+πn,x=7π12+πn.

Since the interval is x[0,π), you get that x{7π12,11π12}. The graph meets the x-axis at the points (7π12,0) and (11π12,0).

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