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MAT 150 Unit 2-2: Solving Quadratic Equations Objectives Solve Solve quadratic equations using factoring 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 Factoring Methods An equation that can be written in the form ax2 + 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. Solve with Factoring A. 𝑥 2 +4𝑥 − 5 = 0 B. 3𝑥 2 + 7𝑥 = 6 C. 3𝑥 2 − 9𝑥 = 0 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 – 16t2 feet, where t is the number of seconds after the ball is thrown. How long will the ball be in the air? Solution Example Consider the daily profit from the production and sale of x units of a product, given by P(x) = –0.01x2 + 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? Example Consider the daily profit from the production and sale of x units of a product, given by P(x) = –0.01x2 + 20x – 500 dollars. Use a graph to find the levels of production and sales that give a daily profit of $1400. Solution a. Example (cont) Consider the daily profit from the production and sale of x units of a product, given by P(x) = –0.01x2 + 20x – 500 dollars. b. Is it possible for the profit to be greater than $1400? Solution 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. The Square Root Method Square Root Method The solutions of the quadratic equation x2 = C are x = C . 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. Example Solve the following equations using the square root method. a. 3x2 – 6 = 0 b. (x – 2)2 = 7 Solution Quadratic Formula The solutions of the quadratic equation ax2 + bx + c = 0 are given by the formula 2 b b 4ac x 2a Note that a is the coefficient of x2, b is the coefficient of x, and c is the constant term. Example Solve 5x2 – 8x = 3 using the quadratic formula. Solution The Discriminant We can also determine the type of solutions a quadratic equation has by looking at the expression b2 4ac, which is called the discriminant of the quadratic equation ax2 + 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 b2 4ac > 0, there are two different real solutions. • If b2 4ac = 0, there is one real solution. • If b2 4ac < 0, there is no real solution. Aids for Solving Quadratic Equations Example Solve the equations. a. x2 = – 36 Solution b. 3x2 + 36 = 0 Example Solve the equations. a. x2 – 3x + 5 = 0 Solution a. b. 3x2 + 4x = –3 Example (cont) Solve the equations. a. x2 – 3x + 5 = 0 Solution b. b. 3x2 + 4x = –3 Market Equilibrium Suppose that the demand for artificial Christmas trees is given by the function 𝑝 = 109.70 − 0.10𝑞 and that the supply of these trees is given by 𝑝 = 0.01 𝑞 2 + 5.91 where p is the price of a tree in dollars and q is the quantity of trees that are demanded/supplied in hundreds. Find the price that gives the market equilibrium price and the number of trees that will be sold/bought at this price.