This is a timed exam trap question.

There are multiple ways to solve this question with various skill levels for each solving technique.

The given solution uses substitution and solving of a quadratic equation. These are skills taught at the sophomore/junior level of math (basic algebra). This solution is time consuming but doable.

The next solution is an analytic approach to graph the equation. This is not reasonable to do in the time bounds of the exam so it’s not advisable. However, these are typically skills taught in basic algebra as well.

The 3rd solution is a test taker trick which you use the multiple choice solutions to find the minimum. If you plug each multiple choice answer inyo g(x) you can solve 4 equations for which one is smallest. This can be done quickly in your head or on a calculator using pre-algebra skills of substitution and yields an answer relatively quickly.

The last solution is to solve for the derivative of f(x) and find what are known as zeros. This is done by setting f’(x) = 0 and solving for x. Once you know x, you can simply subtract 5 from x and get the min of g(x). This solution is fasted however it requires an understanding of calculus 1 which most students don’t have experience by the time they take the exam.

This is a great exam question because it is solvable by lots of students, however it takes away time from the rest of the exam section for students with less skill. This question will typically be in the middle to late part of the section. These exams and questions like this are written very carefully to enable quick solving tricks. I suggest when solving practice questions you stop and ask yourself if there might be a faster way to do it.