Name: Block: Unit : Part 3 Solving Quadratics Day 1 Day Day 3 Day Day 5 Day 6 Day 7 Factoring Zero Product Property Small Quiz: Factoring & Solving Quadratic Formula (QF) Completing the Square (CTS) Review: Solving Quadratics by Factoring, QF, CTS Quiz: Unit Part 3 Solving Quadratics
Tentative Schedule of Upcoming Classes Day 1 Day Day 3 Day Day 5 Day 6 Day 7 A Mon 11/ 30 Factoring B Tues 1/1 A Wed 1/ Zero Product Property B Thurs 1/3 A Fri 1/ Quiz: Factoring & Solving B Mon 1/7 Skills Review Assigned A Tues 1/8 Quadratic Formula B Wed 1/9 A Thurs 1/10 Completing the Square B Fri 1/11 A Mon 1/1 Review B Tues 1/15 Skills Check: Skills Review # A Wed 1/16 Quiz B Thurs 1/17 Absent? See Ms. Huelsman AS SOON AS POSSIBLE to get work and any help you need. Notes are always posted online on the calendar. (If links are not cooperative, try changing to list mode) Handouts and homework keys are posted under assignments You may also email Ms. Huelsman at Kelsey.huelsman@lcps.org with any questions! Need Help? Ms. Huelsman and Mu Alpha Theta are available to help Monday, Tuesday, Thursday, and Friday mornings in L506 starting at 8:10. Ms. Huelsman is in L0 on Wednesday mornings. Need to make up a test/quiz? Math Make Up Room schedule is posted around the math hallway & in Ms. Huelsman s classroom
Day 1 Notes: Factoring Review Let s review factoring techniques for quadratic functions so that we can find the x-intercepts of a quadratic function without graphing. Discuss: What are some things we know about the QUADRATIC function family? How did we find key parts of a Quadratic? 1. Vertex 3 y 1. Y-intercept 3. Increasing and decreasing intervals. End behavior 5. Zero s (solutions, roots, x-intercepts) x 3 1 1 3 5 6 1 3 WHY FACTOR quadratics? To find the zero s analytically 1. Factoring quadratic TRINOMIAL with leading coefficient of 1 (a = 1) Quadratic Term x + bx + c Constant Linear Term Steps to factoring: T 1. Decide the signs for the parentheses based on the CONSTANT TERM. c is positive same signs c is negative different signs. Find # s that MULTIPLY to the constant term and ADD to the linear term 3. Write as two binomials. (note: The LARGER # from step should go with the sign of the linear term.). Check your answer (FOIL)
Let s practice: 1. x 1x + 3. s + 5s 36 Example: x 16x + 6 = A SPECIAL TRINOMIAL - Perfect Square Trinomials: We can summarize this pattern: a + ab + b = (a + b)(a + b) = (a + b) a - ab + b = (a - b)(a - b) = (a - b) 1. x + 10x + 5 =. x + 1x + 9 = Example: x 5 = x + 0x 5 = A SPECIAL BINOMIAL - Difference of two squares: We can summarize this pattern: a b : (a + b)(a b) 1. x 9 =. 9x 11 = 3. x + 100 = What if the leading coefficient is NOT 1? Factoring out a Greatest Common Factor 1. -x + 196. x + 13x 3 + 30x
What if we don t have a GCF and we re stuck with a coefficient that isn t 1? Factoring quadratic TRINOMIAL with leading coefficient 1 Leading Coefficient Ax + Bx + C The leading coefficient is. Quadratic Term Linear Term Constant Steps for factoring trinomials by grouping 1. Decide your signs for the parentheses. Suggestion put your negative first.. Multiply A C 3. Find # s that multiply to equal A C and add to the linear term (B).. Rewrite Bx as a sum of the two factors. There will be terms. 5. Factor by grouping: Group the first two terms and the last two terms Factor the GCF out of each group {the leftovers in parentheses should match} Use distributive property to write as two binomial 6. Check your answer FOIL!! Example: Factor x + 7x + 6 Step 1: Decide signs (neg first) Step : Mult A * C Step 3: M A* C / A B Step : Rewrite Bx as sum of factors Step 5: Factor by Grouping Step 6: FOIL to check
Factor 1. 15 + 13x + x. 1a 31a + 7 7 y y +. 9x 1x 5 3. 18 8 5. 6x 5x + 6. 3x 18x+ 15
Factoring Practice: x 5 (x + 1) 5x 15 3x + 18x - 8 x + x + 1 x + 5x + 3 x x + 1 x 6 y x 1x + 9 6y 13y 5 X + 9 X + 100
Day Notes: Using the Zero Product Property Let s learn and apply the zero product property to solve a quadratic equation so that we can find the solutions (x-intercepts) of quadratics that can't be solved with the square root method. Key Ideas About SOLUTIONS to Quadratic Equations Quadratics can have 1,, or 0 REAL solutions. The REAL solution is the x-coordinate where the parabola crosses the x-axis. The y-coordinate of ANY point on the x-axis is 0, so. to find solutions, we set our quadratic = 0. We've solved quadratics using the SQUARE ROOT METHOD 1. x 10 = 0. Solve the equation (x 6) = (Why can you use the square root method on this?) 10 y 8 6 x 10 8 6 6 8 10 6 8 10 10 y 8 6 x 10 8 6 6 8 10 6 8 10 BUT WHAT IF we are stuck with an x term? We must solve the quadratic BY FACTORING and USING THE ZERO PRODUCT PROPERTY Zero Product Property: If the product of two expressions is zero then one or both of the expressions must be zero. If A B = 0, then either A = 0 or B = 0
We can apply this concept to quadratic equations to find the solutions: If x 6x 5 = 0 (x 1)(x 5) = 0 Then either (x 1) = 0 or (x 5) = 0 Notice that the FACTORS and the SOLUTIONS switch signs! Therefore x = or x = (Graph this to check the intercepts) 1. Solve the quadratic equation x 6x = 0 (Discuss: How is this different from our example?) 10 y 8 6 x 10 8 6 6 8 10 6 8 10. Solve the quadratic equation x x = 5 10 y 8 6 x 10 8 6 6 8 10 6 8 10 Discuss: How is using the Zero Product Property to solve a quadratic equation related to the intercept form of a quadratic function?
3. Find the solutions of x - 11x + 1 = 0. Find the solutions to the quadratic equation x 8x 3 = 3 5x 5. Find the x-intercepts of f(x) = x 1x 10 (What must you do first?) 6. Solve the equation 6x 13x + 3 = 3
Day Notes: The Quadratic Formula Let s learn a new method of solving quadratics the quadratic formula so that we can solve a quadratic equation even when it won't factor. Part 1: The Discriminant How can we tell quickly whether a quadratic has 0, 1, or real solutions? Is there an analytic way to determine this? 1. f(x) = x x. f(x) = x + x y 3 1 1 1 3 1 1 3 1 1 3 1 3 a = b = c = a = b = c = Now calculate: (b) (a)(c) = Now calculate: (b) (a)(c) = ** Discuss: Why are the parentheses important? 3. f(x) = x + x - 1. F(x) = x x + 1 y 3 1 1 3 1 3 3 1 3 1 1 3 1 a = b = c = a = b = c = Now calculate: (b) (a)(c) = Now calculate: (b) (a)(c) = Discuss: What are your observations?
The DISCRIMINANT (b) (a)(c) : if DISCRIMINANT > 0, then DISCRIMINANT DISCRIMINANT = 0, then < 0, then Why do I even need that if I can use my calculator? Try this one -- How many real solutions for y = x + x 13? Where does this come from? 1. Describe the nature of the solution set and determine the solutions to: x 5x = Get this ready: Can this be factored? a. DESCRIBE the nature of the solutions (or solution set) a = b = c = Discriminant: (b) (a)(c) = b. SOLVE the quadratic Use the quadratic formula: x =
. Describe the nature of the solution set and determine the solutions to: x + 10x = -10x - 5 Get this ready: Can this be factored? a. DESCRIBE the solutions (or solution set) a = b = c = Discriminant: (b) (a)(c) = b. SOLVE the quadratic Use the quadratic formula: x = 3. Describe the nature of the solution set and determine the solutions to: x - 6x = -10 Get this ready: Can this be factored? a. DESCRIBE the solutions (or solution set) a = b = c = Discriminant: (b) (a)(c) = b. SOLVE the quadratic Use the quadratic formula: x =
How about these? Can you SOLVE these quadratics? Discuss: Could you use factoring? 1. x + x + 1 = 0. x + x 1 = 0 3. 3x x = -5
Day 5 Notes: Completing the Square Let s learn ANOTHER method of solving quadratics to use when we can t factor so that later, we can find the vertex and center of important shapes. We ve been having FUN with quadratics SOLVING them, & GRAPHING them. 1. Solve & Graph y = x x - 1 10 y 8 6 x 10 8 6 6 8 10 6 8 10. Solve & Graph y = (x )(x + ) 10 y 8 6 x 10 8 6 6 8 10 6 8 10 3. y = x + 10x + 5 Solve it: Graph it: 10 y 8 6 x 10 8 6 6 8 10 6 8 10
Dealing with perfect square trinomials is NICE! Our next method takes advantage of this and FORCES quadratics into being a perfect square trinomial. Background for Completing the Square Making a perfect square trinomial Trinomial = ax + bx + c Constant term (or c) = Fill in c to make these quadratics a perfect square trinomial y = x x + y = x + x + b y = x 6x + y = x + 6x + y = x 8x + y = x + 8x + How were these similar? How are they different? Steps to Solving Quadratics by COMPLETING THE SQUARE 1. Move constant to one side of the equation. Calculate the c needed to complete the square 3. Add this value to BOTH sides. Rewrite your trinomial as a perfect square 5. Use the square root method to solve the equation 1. x 10x + 1 = 0 10 y 8 6 x 10 8 6 6 8 10 6 8 10
Discuss: How is completing the square related to the vertex form of a quadratic?. x 10x = 10 y 8 6 x 10 8 6 6 8 10 6 8 10 SOLVE this quadratic by completing the square. Note COULD we factor these? 3. x 1x + = 0 Where is the vertex?. x x + 8 = 0 Where is the vertex? Will this parabola cross the x-axis? Why? 5. x + 6x + = 0 Where is the vertex?
How to: Solve by Completing the Square when a 1 3x 36x + 150 = 0 Divide each side (every term) by the coefficient of x YOU TRY. Solve by completing the square. 1. x + 8x + 1 = 0 Discuss: Why would you use the method of. Give an example of a quadratic for each method. 1. FACTORING. SQUARE ROOT METHOD 3. COMPLETING THE SQUARE What is your favorite method of solving a quadratic and WHY?
Factor completely. Review: Solving Quadratics Putting It All Together 1. 9x 5. x x 15 3. 3x +5x 1. 9x 30x + 5 5. 8x 18 6. 5x 30x + 5 7. Give 3 synonyms for solution :,, Our QUADRATIC TOOLBOX We know ways to solve quadratic equations. List them. Explain each technique and WHEN you would use it. 1.. 3..
Using the QUADRATIC TOOLBOX Solve the following using TWO techniques. Specify each one & explain why you chose it. 1. x 7x + 1 = 0 Method 1: Reason:. x = 1 - x Method 1: Reason: Work: Work: Method : Reason: Method : Reason: Work: Work:
3. x = 3x + 15 Method 1: Reason:. x + 8x 13 = 0 Method 1: Reason: Work: Work: Method : Reason: Method : Reason: Work: Work:
For the quadratic x x 8 = 0 Solve by the Zero Product Property (Factoring) x x 8 = 0 Solve by Completing the Square x x 8 = 0 Solve Using the Quadratic Formula x x 8 = 0 Solutions: Rewrite to Intercept Form y = x x 8 Solutions: Rewrite to Vertex Form y = x x 8 Solutions: Describe the Root Find the Discriminant: Intercept form: Identify the x-intercepts (, ) and (, ) Vertex Form: Identify the Vertex: Identify the Number and Type of Solutions: Given y = x x 8 Graph: y = x x 8 Find the Vertex: Axis of symmetry: 10 8 6 y x Direction: Size (Vertical Stretch, Vertical Shrink or Standard): 10 8 6 6 8 10 6 8 10