Discriminant and Quadratic 9.5 Day 1 Discriminant and Quadratic

Warm-Up Evaluate the expression (𝑏) 2 −4(𝑎)(𝑐) for the given values. 1. 𝑎=2, 𝑏=8, 𝑐=1 2. 𝑎=3, 𝑏=6, 𝑐=3 3. 𝑎=5,𝑏=3,𝑐=6

Discriminant  

What does the Discriminant Determine? (𝑏) 2 −4(𝑎)(𝑐) = any positive number (𝑏) 2 −4(𝑎)(𝑐) = 0 (𝑏) 2 −4(𝑎)(𝑐) = any negative number Number and Type of Solutions   Number of x-intercepts

Example 1 Find the value of the discriminant and use the value to tell if the equation has two solutions, one solution, or no solution. a) x2 – 2x + 4 = 0 b) –3x2 + 5x – 1 = 0 c) –x2 – 10x – 25 = 0

Example 2 a) y = x2 + 6x + 3 b) y = x2 + 6x + 10 c) y = x2 + 6x + 9 Use the related equation to find the number of x-intercepts of the graph of the function. Then match the equation to the graph. a) y = x2 + 6x + 3 b) y = x2 + 6x + 10 c) y = x2 + 6x + 9

Quadratic Formula The solutions of the quadratic equation ax2 + bx + c = 0 where a ≠ 0 are given by the QUADRATIC FORMULA:   Steps to Solve: Get the quadratic equation in form Evaluate for the - how many and what type of solution do you have? Factor or continue quadratic formula.

Example 1 a) Solve x2 + 5x – 6 = 0 using the quadratic formula.

Example 1 cont. b) Solve: −3 𝑥 2 +𝑥=−5 using the quadratic formula.  

Example 1 cont. c) Solve: 8𝑥 2 −5𝑥=−2 using the quadratic formula.

Example 1 cont. d) Solve 𝑥 2 +8𝑥=−16 using the quadratic formula.

Summary 1. Find the x-intercepts of the graph of y = x2 + 3x – 8.  Remember what we learned in 9.2? What do we know about the x-intercepts of a quadratic function?. . . The x-intercepts occur when y = 0, they are solutions to the quadratic function. So we can now use the quadratic formula.   2. Find the x-intercept of the graph of 4x2 – x – 7 = y.