Identify all of the roots of [tex]f(x)=x^2+2 x+2[/tex]
A. [tex]x=-2+i[/tex] and [tex]x=-2-i[/tex]
B. [tex]x=-1[/tex] and [tex]x=-2[/tex]
C. [tex]x=1+i[/tex] and [tex]x=1-i[/tex]
D. [tex]x=-1+i[/tex] and [tex]x=-1-j[/tex]
Answer (1)
Identify the coefficients: a = 1 , b = 2 , c = 2 .
Apply the quadratic formula: x = 2 a − b ± b 2 − 4 a c .
Substitute and simplify: x = 2 − 2 ± − 4 = 2 − 2 ± 2 i .
Determine the roots: x = − 1 + i and x = − 1 − i . The final answer is x = − 1 ± i .
Explanation
Understanding the Problem We are given the quadratic function f ( x ) = x 2 + 2 x + 2 . Our goal is to find the roots of this function, which are the values of x that make f ( x ) = 0 . We can use the quadratic formula to solve for these roots.
Applying the Quadratic Formula The quadratic formula is given by: x = 2 a − b ± b 2 − 4 a c where a , b , and c are the coefficients of the quadratic equation a x 2 + b x + c = 0 . In our case, a = 1 , b = 2 , and c = 2 .
Substituting the Values Substitute the values of a , b , and c into the quadratic formula: x = 2 ( 1 ) − 2 ± 2 2 − 4 ( 1 ) ( 2 )
Simplifying the Expression Simplify the expression: x = 2 − 2 ± 4 − 8 = 2 − 2 ± − 4 Since − 4 = 2 i , we have: x = 2 − 2 ± 2 i Divide both terms in the numerator by 2: x = − 1 ± i
Finding the Roots Therefore, the roots are x = − 1 + i and x = − 1 − i .
Final Answer The roots of the quadratic function f ( x ) = x 2 + 2 x + 2 are x = − 1 + i and x = − 1 − i .
Examples
Complex numbers and quadratic equations might seem abstract, but they're incredibly useful in electrical engineering. For example, when analyzing AC circuits, impedance (a measure of opposition to current) is often expressed as a complex number. Solving quadratic equations with complex roots helps engineers understand the behavior of these circuits, ensuring stable and efficient designs. This knowledge is crucial for designing everything from power grids to smartphones.
