The Quadratic Formula and the Discriminant. Completing the Square. The Quadratic Formula and the Discriminant. What you’ll learn To solve quadratic equations by completing the square. To solve quadratic equations using the quadratic formula. To find the number of solutions of a quadratic equation. Vocabulary Completing the square. Quadratic formula. Discriminant.

Take a note: A very important note: In general, you can change the expression into a perfect square trinomial by adding This process is called completing the square. The process is the same whether b is positive or negative. Example:

Problem 1: Solving What are the solutions of the equation ? Your turn Answer -13,19

Problem 2: Solving What are the solutions of the equation ? Your turn Answer -2.21,-6.79

Take a note: You can find the solution(s) of any quadratic equation using the quadratic formula 𝑥= −𝑏± 𝑏 2 −4𝑎𝑐 2𝑎 Example: 𝑥= −3± 3 2 −4(2)(−5) 2(2) = −3± 49 4 = −3±7 4 𝑥= −3−7 4 = −10 4 𝑥= −3+7 4 = 4 4 𝑥=− 5 2 𝑥=1

Problem 3: Using the Quadratic Formula What are the solutions of .Use the QF Make the equation =0 𝑥= −𝑏± 𝑏 2 −4𝑎𝑐 2𝑎 𝑥= −(−2)± (−2) 2 −4(1)(−8) 2(1) = 4± 36 2

What are the solutions of .Use the QF Your turn What are the solutions of .Use the QF Answers: a)-3,7 b) -2,5 Your turn again: Which method would you choose to solve each equation? Answers

Problem 4: Finding Approximate Solutions. In the shot put, an athlete throws a heavy metal ball through the air. The arc of the ball can be model by the equation where x is the horizontal distance, in meters, from the athlete and y is the height, in meters of the ball. How far from the athlete will the ball land? 𝑥= −𝑏± 𝑏 2 −4𝑎𝑐 2𝑎 𝑥= −0.84± 0.84 2 −4(−0.04)(2) 2(−0.04)

Your turn Answer: 144.8 ft A batter strikes a baseball. The equation models its path, where x is the horizontal distance in ft., the ball travels and y is the height, in feet, of the ball. How far from the batter will the ball land? Rounded to nearest tenth of a foot. Answer: 144.8 ft

𝑥= −𝑏± 𝑏 2 −4𝑎𝑐 2𝑎 Take a note: In this formula

Problem 5: Using the Discriminant How many real solutions does Because the discriminant is negative has no real solutions Your turn Answer: 2

Classwork odd Homework even TB pgs. 564-565 exercises 7-30 and 34-42 pgs. 571-572 exercises 7-40