1. f(x) = x^2 - 4
2. g(x) = x^2 - x - 12
Factor the quadratic expression completely, and find the roots of the expression.
3. 20x^2 + 13x + 2
4. 49x^2 - 14x - 3
Complete the square, and find the roots of the quadratic equation.
5. x^2 + 16x = 0
In questions 6-10, use the discriminant to determine the number of solutions of the quadratic equation, and whether the solutions are real or complex. Note: It is not necessary to find the roots; just determine the number and types of solutions.
6. x^2 + 6x - 7 = 0
7. z^2 + z + 1 = 0
8. (3)1/2y^2 - 4y - 7(3)1/2 = 0
9. 2x^2 - 10x + 25 = 0
10. 2x2 - 6x + 5 = 0
11. Steve traveled 200 miles at a certain speed. Had he gone 10mph faster, the trip would have taken 1 hour less. Find the speed of his vehicle.
12. The Hudson River flows at a rate of 3 miles per hour. A patrol boat travels 60 miles upriver, and returns in a total time of 9 hours. What is the speed of the boat in still water?
13. A designer attempts to arrange the characters of his artwork in the form of a square grid with equal numbers of rows and columns, but finds that 24 characters are left out. When he tries to add one more row and column, he finds that he has 25 too few characters. Find the number of characters used by the designer.
This solution is comprised of detailed step-by-step calculation and explanation of the given problems. The solution also provides students with a clear perspective of the underlying mathematical concepts.