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Chapter 16 Quadratic Equations

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Martin-Gay, Developmental Mathematics 2 16.1 – Solving Quadratic Equations by the Square Root Property 16.2 – Solving Quadratic Equations by Completing the Square 16.3 – Solving Quadratic Equations by the Quadratic Formula 16.4 – Graphing Quadratic Equations in Two Variables 16.5 – Interval Notation, Finding Domains and Ranges from Graphs and Graphing Piecewise-Defined Functions Chapter Sections

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§ 16.1 Solving Quadratic Equations by the Square Root Property

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Martin-Gay, Developmental Mathematics 4 Square Root Property We previously have used factoring to solve quadratic equations. This chapter will introduce additional methods for solving quadratic equations. Square Root Property If b is a real number and a 2 = b, then

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Martin-Gay, Developmental Mathematics 5 Solve x 2 = 49 Solve (y – 3) 2 = 4 Solve 2x 2 = 4 x 2 = 2 y = 3 2 y = 1 or 5 Square Root Property Example

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Martin-Gay, Developmental Mathematics 6 Solve x 2 + 4 = 0 x 2 = 4 There is no real solution because the square root of 4 is not a real number. Square Root Property Example

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Martin-Gay, Developmental Mathematics 7 Solve (x + 2) 2 = 25 x = 2 ± 5 x = 2 + 5 or x = 2 – 5 x = 3 or x = 7 Square Root Property Example

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Martin-Gay, Developmental Mathematics 8 Solve (3x – 17) 2 = 28 3x – 17 = Square Root Property Example

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§ 16.2 Solving Quadratic Equations by Completing the Square

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Martin-Gay, Developmental Mathematics 10 In all four of the previous examples, the constant in the square on the right side, is half the coefficient of the x term on the left. Also, the constant on the left is the square of the constant on the right. So, to find the constant term of a perfect square trinomial, we need to take the square of half the coefficient of the x term in the trinomial (as long as the coefficient of the x 2 term is 1, as in our previous examples). Completing the Square

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Martin-Gay, Developmental Mathematics 11 What constant term should be added to the following expressions to create a perfect square trinomial? x 2 – 10x add 5 2 = 25 x 2 + 16x add 8 2 = 64 x 2 – 7x add Completing the Square Example

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Martin-Gay, Developmental Mathematics 12 We now look at a method for solving quadratics that involves a technique called completing the square. It involves creating a trinomial that is a perfect square, setting the factored trinomial equal to a constant, then using the square root property from the previous section. Completing the Square Example

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Martin-Gay, Developmental Mathematics 13 Solving a Quadratic Equation by Completing a Square 1)If the coefficient of x 2 is NOT 1, divide both sides of the equation by the coefficient. 2)Isolate all variable terms on one side of the equation. 3)Complete the square (half the coefficient of the x term squared, added to both sides of the equation). 4)Factor the resulting trinomial. 5)Use the square root property. Completing the Square

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Martin-Gay, Developmental Mathematics 14 Solve by completing the square. y 2 + 6y = 8 y 2 + 6y + 9 = 8 + 9 (y + 3) 2 = 1 y = 3 ± 1 y = 4 or 2 y + 3 = ± = ± 1 Solving Equations Example

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Martin-Gay, Developmental Mathematics 15 Solve by completing the square. y 2 + y – 7 = 0 y 2 + y = 7 y 2 + y + ¼ = 7 + ¼ (y + ½) 2 = Solving Equations Example

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Martin-Gay, Developmental Mathematics 16 Solve by completing the square. 2x 2 + 14x – 1 = 0 2x 2 + 14x = 1 x 2 + 7x = ½ x 2 + 7x + = ½ + = (x + ) 2 = Solving Equations Example

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§ 16.3 Solving Quadratic Equations by the Quadratic Formula

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Martin-Gay, Developmental Mathematics 18 The Quadratic Formula Another technique for solving quadratic equations is to use the quadratic formula. The formula is derived from completing the square of a general quadratic equation.

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Martin-Gay, Developmental Mathematics 19 A quadratic equation written in standard form, ax 2 + bx + c = 0, has the solutions. The Quadratic Formula

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Martin-Gay, Developmental Mathematics 20 Solve 11n 2 – 9n = 1 by the quadratic formula. 11n 2 – 9n – 1 = 0, so a = 11, b = -9, c = -1 The Quadratic Formula Example

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Martin-Gay, Developmental Mathematics 21 x 2 + 8x – 20 = 0 (multiply both sides by 8) a = 1, b = 8, c = 20 Solve x 2 + x – = 0 by the quadratic formula. The Quadratic Formula Example

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Martin-Gay, Developmental Mathematics 22 Solve x(x + 6) = 30 by the quadratic formula. x 2 + 6x + 30 = 0 a = 1, b = 6, c = 30 So there is no real solution. The Quadratic Formula Example

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Martin-Gay, Developmental Mathematics 23 The expression under the radical sign in the formula (b 2 – 4ac) is called the discriminant. The discriminant will take on a value that is positive, 0, or negative. The value of the discriminant indicates two distinct real solutions, one real solution, or no real solutions, respectively. The Discriminant

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Martin-Gay, Developmental Mathematics 24 Use the discriminant to determine the number and type of solutions for the following equation. 5 – 4x + 12x 2 = 0 a = 12, b = –4, and c = 5 b 2 – 4ac = (–4) 2 – 4(12)(5) = 16 – 240 = –224 There are no real solutions. The Discriminant Example

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Martin-Gay, Developmental Mathematics 25 Solving Quadratic Equations Steps in Solving Quadratic Equations 1)If the equation is in the form (ax+b) 2 = c, use the square root property to solve. 2)If not solved in step 1, write the equation in standard form. 3)Try to solve by factoring. 4)If you haven’t solved it yet, use the quadratic formula.

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Martin-Gay, Developmental Mathematics 26 Solve 12x = 4x 2 + 4. 0 = 4x 2 – 12x + 4 0 = 4(x 2 – 3x + 1) Let a = 1, b = -3, c = 1 Solving Equations Example

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Martin-Gay, Developmental Mathematics 27 Solve the following quadratic equation. Solving Equations Example

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§ 16.4 Graphing Quadratic Equations in Two Variables

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Martin-Gay, Developmental Mathematics 29 We spent a lot of time graphing linear equations in chapter 3. The graph of a quadratic equation is a parabola. The highest point or lowest point on the parabola is the vertex. Axis of symmetry is the line that runs through the vertex and through the middle of the parabola. Graphs of Quadratic Equations

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Martin-Gay, Developmental Mathematics 30 x y Graph y = 2x 2 – 4. x y 0 –4–4 1 –2–2 –1–1 –2–2 24 –2–24 (2, 4) (–2, 4) (1, –2)(–1, – 2) (0, –4) Graphs of Quadratic Equations Example

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Martin-Gay, Developmental Mathematics 31 Although we can simply plot points, it is helpful to know some information about the parabola we will be graphing prior to finding individual points. To find x-intercepts of the parabola, let y = 0 and solve for x. To find y-intercepts of the parabola, let x = 0 and solve for y. Intercepts of the Parabola

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Martin-Gay, Developmental Mathematics 32 If the quadratic equation is written in standard form, y = ax 2 + bx + c, 1) the parabola opens up when a > 0 and opens down when a < 0. 2) the x-coordinate of the vertex is. To find the corresponding y-coordinate, you substitute the x-coordinate into the equation and evaluate for y. Characteristics of the Parabola

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Martin-Gay, Developmental Mathematics 33 x y Graph y = – 2x 2 + 4x + 5. x y 1 7 2 5 05 3–1 (3, –1) (–1, –1) (2, 5) (0, 5) (1, 7) Since a = –2 and b = 4, the graph opens down and the x-coordinate of the vertex is Graphs of Quadratic Equations Example

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§ 16.5 Interval Notation, Finding Domain and Ranges from Graphs, and Graphing Piecewise-Defined Functions

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Martin-Gay, Developmental Mathematics 35 Recall that a set of ordered pairs is also called a relation. The domain is the set of x-coordinates of the ordered pairs. The range is the set of y-coordinates of the ordered pairs. Domain and Range

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Martin-Gay, Developmental Mathematics 36 Find the domain and range of the relation {(4,9), (–4,9), (2,3), (10, –5)} Domain is the set of all x-values, {4, –4, 2, 10} Range is the set of all y-values, {9, 3, –5} Example Domain and Range

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Martin-Gay, Developmental Mathematics 37 Find the domain and range of the function graphed to the right. Use interval notation. x y Domain is [ – 3, 4] Domain Range is [ – 4, 2] Range Example Domain and Range

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Martin-Gay, Developmental Mathematics 38 Find the domain and range of the function graphed to the right. Use interval notation. x y Domain is (– , ) Domain Range is [– 2, ) Range Example Domain and Range

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Martin-Gay, Developmental Mathematics 39 Input (Animal) Polar Bear Cow Chimpanzee Giraffe Gorilla Kangaroo Red Fox Output (Life Span) 20 15 10 7 Find the domain and range of the following relation. Example Domain and Range

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Martin-Gay, Developmental Mathematics 40 Domain is {Polar Bear, Cow, Chimpanzee, Giraffe, Gorilla, Kangaroo, Red Fox} Range is {20, 15, 10, 7} Domain and Range Example continued

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Martin-Gay, Developmental Mathematics 41 Graph each “piece” separately. Graph Graphing Piecewise-Defined Functions Example Continued. x f (x) = 3x – 1 0– 1 (closed circle) –1– 4 –2– 7 x f (x) = x + 3 1 4 2 5 3 6 Values 0. Values > 0.

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Martin-Gay, Developmental Mathematics 42 Example continued Graphing Piecewise-Defined Functions x y x f (x) = x + 3 1 4 2 5 3 6 x f (x) = 3x – 1 0– 1 (closed circle) –1– 4 –2– 7 (0, –1) (–1, 4) (–2, 7) Open circle (0, 3) (3, 6)

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