Final answer:
All listed methods of solving quadratic equations can yield complex roots if those roots exist, except for factoring, which requires that real factors be feasibly found. Completing the square, the quadratic formula, and graphical methods can all reveal complex roots, as these involve the potential for taking square roots of negative numbers.
Step-by-step explanation:
The question explores the various methods of solving quadratic equations and asks which of them do not yield complex roots. Quadratic equations can sometimes have complex roots, which are not real number solutions but rather have an imaginary component (i.e., involving the square root of a negative number). However, when we focus on quadratic equations derived from physical data, real roots are generally expected; complex solutions are less likely to occur in those contexts.
Each of the solving methods listed can potentially yield complex roots if the quadratic equation has no real roots. However, if we are considering only situations where complex roots are not expected:
- A) Factoring - This method starts with the premise that the quadratic can be broken down into a product of linear factors. If the quadratic is factorable over the real numbers, the solutions will be real.
- B) Completing the square and C) Quadratic formula - Both of these methods involve potentially taking the square root of a number. If the number under the square root (the discriminant) is negative, these methods yield complex roots. However, when solving equations based on physical data, the discriminant is typically non-negative, which means real roots are calculated.
- D) Graphical method - If the graph of the quadratic equation crosses the x-axis, it has real roots which can be estimated from the graph. However, if the graph does not intersect the x-axis, this method suggests that the roots are complex.
Of these methods, the only one that inherently does not yield complex roots when they exist is factoring, as it requires the quadratic to be expressible in real factors to begin with. The other methods, including Two-Dimensional (x-y) Graphing, can reveal complex roots if they exist.
In some types of equilibrium problems, being able to solve for square roots, which are a pivotal part of quadratic solutions, can be necessary to determine a final answer.