CCSS: N.CN.7: Solve quadratic equations with real coefficients that have complex solutions

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1 4.5 Completing The Square 1) Solve quadratic equations using the square root property 2) Generate perfect square trinomials by completing the square 3) Solve quadratic equations by completing the square CCSS: N.CN.7: Solve quadratic equations with real coefficients that have complex solutions F.IF.8.a: Use the process of factoring and completing the square in a quadratic function to show zeros, extreme values, and symmetry of the graph, and interpret these in terms of a context 1. Look for and make use of structure 1. The square root property: a. Powers and roots are inverses, thus raising to a power and taking a root are inverse operations b. If the quadratic term in an equation can be isolated, it can be solved by taking the square root of each side c. Works best in an equation that does not have a linear term. d. Can be applied to perfect square trinomials because they can be written as a binomial raised to the second power 2. To generate a perfect square trinomial (complete the square): a. x 2 + bx + c; c = (0.5b) 2 b. so x 2 + bx + (b/2) 2 will always be a perfect square 3. To solve a quadratic equation by completing the square, the leading coefficient must be Example 1: Solve x x + 49 = 64 by using the square root property. 5. Example 2: Solve x 2-4x + 4 = 13 by using the square root property. 6. Example 3: Find the value of c that makes x x + c a perfect square. Then write the trinomial as a perfect square. 7. Example 4: Solve x 2 + 4x 12 = 0 by completing the square. 8. Example 5: Solve 3x 2-2x 1 = 0 by completing the square. 9. Example 6: Solve x 2 + 4x + 11 = 0 by completing the square

2 4.6 The Quadratic Formula and the Discriminant 1) solve quadratic equations using the quadratic formula 2) Use the discriminant to determine the number and type of roots of a quadratic equation CCSS: N.CN.7: Solve quadratic equations with real coefficients that have complex solutions A.SSE.1.b: Interpret complicated expressions by viewing one or more of their parts as a single entity 2. Look for and express regularity in repeated reasoning The Quadratic Formula uses the "a", "b", and "c" from "ax 2 + bx + c", where "a", "b", and "c" are just numbers; they are the "numerical coefficients". The Formula is derived from the process of completing the square, and is formally stated as: For ax 2 + bx + c = 0, the value of x is given by: Here are some examples of how the Quadratic Formula works: Solve x 2 + 3x 4 = 0 This quadratic happens to factor: x 2 + 3x 4 = (x + 4)(x 1) = 0...so I already know that the solutions are x = 4 and x = 1. How would my solution look in the Quadratic Formula? Using a = 1, b = 3, and c = 4, my solution looks like this: Then, as expected, the solution is x = 4, x = 1. Source:

3 1. All quadratic equations have two solutions. 2. If both solutions are real and different, the parabola has two x-intercepts. 3. If it has only one real solution, the quadratic is a perfect square. a. It still has two roots, they just happen to be the same number. b. The parabola will touch the x-axis only once. c. The double root will also be the vertex of the parabola. 4. The equation has no real solutions if a. It has a minimum that is above the x-axis or b. It has a maximum that is below the x-axis c. Both of its solutions are complex. 5. The discriminant: D=b 2 4ac (the radicand in the quadratic formula) can be used to determine the number of real roots. 6. If D>0 and is a perfect square, will have 2 real, rational roots. 7. If D>0 and is not a perfect square, will have 2 real, irrational roots. 8. If D=0, will have one real, rational root. ( b/2a,0) will be both roots and the x- intercept. 9. If D<0, will have no real (2 complex) roots. 10. Summary of methods for solving quadratic equations. a. Graphing (4.1, 4.2) b. Factoring (4.3) c. Square Root Property (4.4) d. Completing the Square (4.4, 4.5) e. Quadratic Formula (4.6) 11. Example 1: Solve x 2-8x = 33 by using the Quadratic Formula. 12. Example 2: Solve x 2-34x = 0 by using the Quadratic Formula. 13. Example 3: Solve x 2-6x + 2 = 0 by using the Quadratic Formula. 14. Example 4: Solve x = 6x by using the Quadratic Formula. 15. Example 5: Find the value of the discriminant for each quadratic equation. Then describe the number and type of roots for the equation. a. x 2 + 3x + 5 = 0 b. x 2-11x + 10 = 0

4 4.7 Transformations of Quadratic Graphs 1) Write quadratic equations in vertex form 2) Transform graphs of quadratic functions 3) Write equations of quadratic functions by interpreting the graph CCSS: F.IF.8.a: Use the process of factoring and completing the square in a quadratic function to show zeros, extreme values, and symmetry of the graph, and interpret these in terms of a context. F.BF.3: Identify the effect on the graph of replacing f(x) by f(x) + k, kf(x), f(kx), and f(x+k) for specific values of k (both positive and negative); find the value of k given the graphs. Experiment with cases and illustrate an explanation of the effects on the graph using technology. 7. Look for and make use of structure 1. Vertex form: f(x) = a(x - h) 2 + k Transformations review (from section 2.7): 2. Translation a. Slide b. Addition 3. Reflection a. Flip b. Multiplication by Dilation a. Stretch/compress (enlarge/shrink) b. Multiplication 5. Horizontal/Vertical inside/outside TRANSFORMATION CHANGE TO PARENT GRAPH Translation f(x + h) Horizontal translation h units left f(x - h) Horizontal translation h units right f(x) - h Vertical translation down h units f(x) + h Vertical translation up h units Reflection -f(x) Vertical reflection (across the x axis) f(-x) Horizontal reflection (across the y-axis) Dilation af(x), a > 1 Vertical stretch by a factor of a

5 6. Example 1: Write each equation in vertex form. a. f(x) = x 2 2x + 4 b. f(x) = -3x 2 18x Example 2: Graph f(x) = -2x 2 + 4x Quadratic Inequalities 1) Graph quadratic inequalities in two variables 2) Solve quadratic inequalities in one variable graphically and algebraically CCSS: A.CED.1: Create equations and inequalities in one variable and use them to solve problems A.CED.3: Represent constraints by equations or inequalities, and by systems of equations and/or inequalities, and interpret solutions as viable or nonviable options in a modeling context 1. Make sense of problems and persevere in solving them 1. Is the parabola solid or dotted? 2. Do you shade the inside or the outside (use a test point)? 3. For which x-values will the parabola lie above (or below) the x-axis? 4. Example 1: Graph f(x) > x 2 3x Example 2: Solve x 2 4x + 3 < 0 by graphing. 6. Example 3: Solve 0-2x 2 6x + 1 by graphing. 7. Example 4: Solve x 2 + x 2 algebraically.

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