Identify the zeros of the function [tex]f(x)=x^2+16[/tex].
A. [tex]x= \pm 16 i[/tex]
B. [tex]x= \pm 4[/tex]
C. [tex]x=-4 i[/tex]
D. [tex]x= \pm 4 i[/tex]
Answer (2)
Set the function equal to zero: x 2 + 16 = 0 .
Isolate x 2 : x 2 = − 16 .
Take the square root of both sides: x = ± − 16 .
Simplify to find the zeros: x = ± 4 i .
Explanation
Understanding the Problem We are asked to find the zeros of the function f ( x ) = x 2 + 16 . This means we need to find the values of x for which f ( x ) = 0 .
Setting up the Equation To find the zeros, we set f ( x ) = 0 and solve for x :
x 2 + 16 = 0
Isolating x 2 Subtract 16 from both sides of the equation: x 2 = − 16
Taking the Square Root Take the square root of both sides: x = ± − 16
Simplifying the Square Root Since we have a negative number under the square root, we will have imaginary solutions. Recall that − 1 = i . So we can rewrite the expression as: x = ± 16 ⋅ − 1 = ± 4 i
Examples
Finding the zeros of a function like f ( x ) = x 2 + 16 might seem abstract, but it's a fundamental concept in many areas of science and engineering. For example, in electrical engineering, the roots of a characteristic equation determine the stability of a circuit. If the roots have imaginary parts, the circuit might oscillate. Similarly, in quantum mechanics, finding the zeros of a wavefunction can tell us about the possible energy levels of a particle. Understanding complex roots helps engineers design stable systems and physicists understand the behavior of subatomic particles.
The zeros of the function f ( x ) = x 2 + 16 are found by solving the equation x 2 + 16 = 0 , which results in x = ± 4 i . Therefore, the correct option is D: x = ± 4 i .
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