1. MAT 150 Unit 2-2: Solving Quadratic Equations

2. Objectives  Solve quadratic equations using factoring  Solve quadratic equations graphically using the x-intercept method and the intersection method  Solve quadratic equations by combining graphical and factoring methods  Solve quadratic equations using the square root method  Solve quadratic equations by completing the square  Solve quadratic equations using the quadratic formula  Solve quadratic equations having complex solutions

3. Factoring Methods An equation that can be written in the form ax 2 + bx + c = 0, with a ≠ 0, is called a quadratic equation. Zero Product Property If the product of two real numbers is 0, then at least one of them must be 0. That is, for real numbers a and b, if the product ab = 0, then either a = 0 or b = 0 or both a and b are equal to 0.

4. Solve with Factoring

5. Example The height above ground of a ball thrown upward at 64 feet per second from the top of an 80-foot-high building is modeled by S(t) = 80 + 64t – 16t 2 feet, where t is the number of seconds after the ball is thrown. How long will the ball be in the air? Solution

6. Example Consider the daily profit from the production and sale of x units of a product, given by P(x) = –0.01x 2 + 20x – 500 dollars. a. Use a graph to find the levels of production and sales that give a daily profit of $1400. b. Is it possible for the profit to be greater than $1400?

7. Example Consider the daily profit from the production and sale of x units of a product, given by P(x) = –0.01x 2 + 20x – 500 dollars. a. Use a graph to find the levels of production and sales that give a daily profit of $1400. Solution

8. Example (cont) Consider the daily profit from the production and sale of x units of a product, given by P(x) = –0.01x 2 + 20x – 500 dollars. b. Is it possible for the profit to be greater than $1400? Solution

9. Combining Graphs and Factoring Factor Theorem The polynomial function f has a factor (x – a) if and only if f(a) = 0. Thus, (x – a) is a factor of f(x) if and only if x = a is a solution to f (x) = 0.

10. The Square Root Method Square Root Method The solutions of the quadratic equation x 2 = C are x = Note that, when we take the square root of both sides, we use a ± symbol because there are both a positive and a negative value that, when squared, give C.

11. Example Solve the following equations using the square root method. a. 3x 2 – 6 = 0b. (x – 2) 2 = 7 Solution

12. Quadratic Formula The solutions of the quadratic equation ax 2 + bx + c = 0 are given by the formula Note that a is the coefficient of x 2, b is the coefficient of x, and c is the constant term.

13. Example Solve 5x 2 – 8x = 3 using the quadratic formula. Solution

14. The Discriminant We can also determine the type of solutions a quadratic equation has by looking at the expression b 2  4ac, which is called the discriminant of the quadratic equation ax 2 + bx + c = 0. The discriminant is the expression inside the radical in the quadratic formula, so it determines if the quantity inside the radical is positive, zero, or negative. If b 2  4ac > 0, there are two different real solutions. If b 2  4ac = 0, there is one real solution. If b 2  4ac < 0, there is no real solution.

15. Aids for Solving Quadratic Equations

16. Example Solve the equations. a. x 2 = – 36b. 3x 2 + 36 = 0 Solution

17. Example Solve the equations. a. x 2 – 3x + 5 = 0b. 3x 2 + 4x = –3 Solution a.

18. Example (cont) Solve the equations. a. x 2 – 3x + 5 = 0b. 3x 2 + 4x = –3 Solution b.

19. Market Equilibrium