9.4/9.5: Solving Quadratic Equations Homework 67: p.529: 2-49 EOO, 53-56 All Homework 68: p.536: 25-69 EOO, 77-8 Odd Objectives Solve Quadratic Equations by Graphing Solve Quadratic Equations by the Quadratic Formula Entry Task: Simplify each radical expression. 96 4 6 4. 45 3 5 2. 26 6 6 5. 2 2 3 3. 72 6 2 6. 8 3 2
Concept: Solutions to an Equation Solution to a Quadratic Equation: Any values for x where: ax 2 + bx + c = 0 is true
Example : Simple Quadratics and Their Graphs Solve and Graph 2 x2 = 8 2 2 x2 = 8 2 x 2 = 6 x = ±4 2 4 2 = 8 2 4 2 = 8 2 (6) = 8
Example : Simple Quadratics and Their Graphs 2 x2 = 8 8 8 2 x2 8 = 0 x 4 3 2 0 2 3 4 2 x2 + 8 y (x, y) 2 4 2 8 0 4,0 2 3 2 8 2 2 2 8 2 2 8 2 0 2 8 2 2 8 2 2 2 8 2 3 2 8 2 4 2 8 6 8 6 0 2, 6 (0, 8) (2, 6) (4,0)
Example : Simple Quadratics and Their Graphs Solution = x-intercept Solution = x-intercept Solution: x = 4 Root: x = 4 Solution: x = 4 Root: x = 4 Vertex/y-intercept
Student Led Example : Parabolas and Their Graphs We ll get to this in a second but first, why we don t rely on hand drawn graphs
Concept: Minimums and Maximums If a parabola opens upward, the vertex is a minimum If a parabola opens downward, the vertex is a maximum
Activity: Solving Quadratics with a Calculator Given the function 2 x2 + 2x 6: A. Find the x-intercepts B. Find the y-intercept C. Find the Vertex D. Find the Axis of Symmetry
Activity: Solving Quadratics with a Calculator Given the function 2 x2 + 2x 6: A. Find the x-intercepts Step : Input the function into Y= Step 2: Press 2 nd and Calc, then choose option 2, Zero
Activity: Solving Quadratics with a Calculator Given the function 2 x2 + 2x 6: A. Find the x-intercepts Step 3: Set the Left and Right Bounds for each of the x-intercepts. Then, guess close to the intercept
Activity: Solving Quadratics with a Calculator Given the function 2 x2 + 2x 6: B. Find the y-intercept IT S THIS. There is nothing more you need to do. It s THIS NUMBER!
Activity: Solving Quadratics with a Calculator Given the function 2 x2 + 2x 6: C. Find the Vertex Step 5: Determine if the function opens up or opens down. Step 6: Press 2 nd and Calc, then choose option either Maximum or Minimum
Activity: Solving Quadratics with a Calculator Given the function 2 x2 + 2x 6: C. Find the Vertex Step 7: Use Left and Right bound and Guess the minimum. It s safe to round Vertex = ( 2, 8) Axis of symmetry
Example 2: Parabolas as Solutions [Discussion Only] You and your buddies developed a sling shot large enough to hurl a softball sized snowball across your neighborhood. An angle of 35 degrees to 65 degrees is found to be your sweet spot for carnage. After tracking the path of a few snowballs, you get two equations: y 35 = 0.00735x 2 + 0.700208x + 6 y 65 = 0.040330x 2 + 2.44507x + 6 Which angle provides the greatest distance? Which provides the greatest height?
Example 2: Parabolas as Solutions 65 35 If a 35 angle reaches 56 feet and a 65 angle reaches 73 feet, what angle would you use to hit that annoying kid that is 60 feet from your launcher?
The Quadratic Formula x = b + b 2 b 2 4ac 4 c 2 a
Example 3: The Quadratic Formula Solve x 2 + 9x + 4 b ± b 2 4ac 2a a =, b = 9, c = 4 9 ± 9 2 4()(4) 2()
Example 3: The Quadratic Formula Solve x 2 + 9x + 4 9 ± 9 2 4()(4) 2() 9 ± 8 56 2 9 ± 25 2
Example 3A: The Quadratic Formula Solve x 2 + 9x + 4 9 ± 25 2 9 ± 5 2 9 + 5 2 = 4 2 9 5 = 4 = 2 = 7 2 2
Example 3A: The Quadratic Formula Solve x 2 + 9x + 4 possible solutions: x = { 7, 2} x 2 + 9x + 4 x 2 + 9x + 4 ( 7) 2 +9( 7) + 4 ( 2) 2 +9( 2) + 4 49 63 + 4 4 8 + 4 4 + 4 4 + 4 0 0
Example 3B: The Quadratic Formula Solve 2x 2 3x = 8 Write in Standard Form: 2x 2 3x 8 = 0 b ± b 2 4ac 2a ( 3) ± ( 3) 2 4(2)( 8) 2(2)
Example 3B: The Quadratic Formula Solve 2x 2 3x = 8 ( 3) ± ( 3) 2 4(2)( 8) 2(2) 3 ± 9 + 64 4 3 ± 73 4
Student Led Example 3: The Quadratic Formula Solve each quadratic function using the quadratic formula A x 2 + 3x = 2 C 2 + x 2 = x x = 2 x = 2, B 3x 2 + 7x 20 D x 2 2x 2 x = 5 3, 4 x = ± 2 3
End of Lesson None. I think SLE 3 was enough work to count as an exit task.