Lesson 9.5 Objective: To solve quadratic equations using the quadratic formula. Quadratic formula: When Then the value of x is… What formula can be used to solve any quadratic equation?

Example: Use the quadratic equation to solve for x. a = 1 b = 9 c = 14

Example: Solve for x a = 1 b = 5 c = -6

Example: Solve for x a = -2 b = 6 c = 9 Simplify Simplified

A ball is thrown upwards with an initial velocity of 90 feet per second from a height of 6 feet. Use the vertical motion model to determine the time it will take the ball to hit the ground. Vertical Motion Model h = height of ground t = time v = initial velocity s = starting height a = -16 b = 90 c = 6

The discriminant is the expression inside the radical in the quadratic formula, b 2 – 4ac. If b 2 – 4ac is positive, then the equation has two solutions. If b 2 – 4ac is zero, then the equation has one solution. If b 2 – 4ac is negative, then the equation has no real solution. Applications of the Discriminant

The discriminant also tells the number of times the parabola crosses the x-axis Positive discriminant: The parabola crosses x-axis twice. Zero discriminant: The parabola crosses x-axis once. Negative discriminant: The parabola never crosses x-axis. Positive Two solutions Zero One solution Negative No solutions

Examples: Find the discriminant and determine the number of solutions 1. a =1 b =-3 c = Two solutions 2.2. a =1 b =2 c =5 4 – 20 –16 No solutions – 16 0 One solution