Find all real and non-real roots of the function [tex]f(x)=x^2+49[/tex].
A) [tex]x=-7,7[/tex]
B) [tex]x=-7 i, 7 i[/tex]
C) [tex]x=-49 i, 49 i[/tex]
D) [tex]x=i+7, i-7[/tex]
Answer (1)
Set the function equal to zero: x 2 + 49 = 0 .
Isolate x 2 : x 2 = − 49 .
Take the square root of both sides: x = ± − 49 .
Simplify to find the roots: x = ± 7 i , so the roots are x = − 7 i , 7 i .
Explanation
Understanding the Problem We are given the function f ( x ) = x 2 + 49 and asked to find its roots, which are the values of x for which f ( x ) = 0 .
Setting up the Equation To find the roots, we set f ( x ) = 0 and solve for x :
x 2 + 49 = 0
Isolating x 2 Subtract 49 from both sides of the equation: x 2 = − 49
Taking the Square Root Take the square root of both sides: x = ± − 49
Simplifying the Square Root Since we have a negative number under the square root, we know the roots will be non-real (imaginary). We can rewrite the square root as: x = ± 49 ⋅ − 1 = ± 49 ⋅ − 1 = ± 7 i
Finding the Roots Thus, the roots are x = 7 i and x = − 7 i .
Final Answer The roots of the function f ( x ) = x 2 + 49 are x = − 7 i and x = 7 i . Therefore, the correct answer is B.
Examples
Understanding roots of equations is crucial in many fields. For example, in physics, finding the roots of a characteristic equation helps determine the stability of a system. In engineering, roots can represent resonant frequencies in circuits or critical points in structural analysis. Moreover, in signal processing, roots are used to analyze the frequency content of signals. Knowing how to find both real and complex roots allows for a comprehensive understanding of the system's behavior.
