Types of Zeros Investigation Given the following quadratic functions, find the zeros: 1. x2 -­‐ 49 = 0 x = 7, x = -­‐7 2. x2 + 16 = 0 3. x2 = 0 No solution x = 0 Discuss with your partner the following questions: 1. When can you expect 2 solutions in a quadratic equation? When it crosses the x-­‐axis twice 2. When can you expect 1 solution in a quadratic equation? When it crosses the x-­‐axis once 3. When can you expect a quadratic to have no solutions? When it doesn’t cross the x-­‐axis Recall from the previous lesson, the quadratic formula: 𝑥=
−𝑏 ± 𝑏 ! − 4𝑎𝑐
2𝑎
This part of the quadratic formula helps us to determine how many solutions a quadratic will have: 𝑏 ! − 4𝑎𝑐 It is called the Discriminant. Because the quadratic formula contains a square root, we can determine the number of solutions based on the discriminant. • If the discriminant is negative, how many solutions will the quadratic have? No solutions • If the discriminant is positive, how many solutions will the quadratic have? Two solutions • If the discriminant is zero, how many solutions will the quadratic have? One solution Given the following quadratics, use the discriminant to determine how many solutions it will have. (Remember to solve a quadratic, you have to set it equal to 0!) 1. x2 – 6x + 11 = 2 one solution 2. 3x2 + 5x = 12 two solutions no solutions 4. x2 – 27 = 0 two solutions no solutions 6. x2 + 4x -­‐1 = 0 two solutions 3. 3x2 + 48 = 0 5. x2 + x + 1 = 0 Given the following graphs of quadratic functions, determine the sign of the discriminant. Discriminant is negative Discriminant is positive Discriminant is 0 Discriminant is 0