Solving Quadratic Equations by Completing the Square

Martin-Gay, Developmental Mathematics 2 Square Root Property We previously have used factoring to solve quadratic equations. If the quadratic is either too difficult or impossible to factor, there are additional methods for solving the quadratic equation. Square Root Property If b is a real number and a 2 = b, then

Martin-Gay, Developmental Mathematics 3  “completing the square” involves creating a perfect square trinomial, factoring the trinomial and setting it equal to a constant, then using the square root property to get our solutions Completing the Square

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

§ 16.3 Solving Quadratic Equations by the Quadratic Formula

Martin-Gay, Developmental Mathematics 6 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.

Martin-Gay, Developmental Mathematics 7 A quadratic equation written in standard form, ax 2 + bx + c = 0, has the solutions. The Quadratic Formula

Martin-Gay, Developmental Mathematics 8 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

Martin-Gay, Developmental Mathematics 9 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

Martin-Gay, Developmental Mathematics 10 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

Martin-Gay, Developmental Mathematics 11 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

Martin-Gay, Developmental Mathematics 12 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

Martin-Gay, Developmental Mathematics 13 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.

Martin-Gay, Developmental Mathematics 14 Solve 12x = 4x = 4x 2 – 12x = 4(x 2 – 3x + 1) Let a = 1, b = -3, c = 1 Solving Equations Example

Martin-Gay, Developmental Mathematics 15 Solve the following quadratic equation. Solving Equations Example

§ 16.4 Graphing Quadratic Equations in Two Variables

Martin-Gay, Developmental Mathematics 17 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

Martin-Gay, Developmental Mathematics 18 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

Martin-Gay, Developmental Mathematics 19 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

Martin-Gay, Developmental Mathematics 20 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

Martin-Gay, Developmental Mathematics 21 x y Graph y = – 2x 2 + 4x + 5. x y –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

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

Martin-Gay, Developmental Mathematics 23 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

Martin-Gay, Developmental Mathematics 24 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

Martin-Gay, Developmental Mathematics 25 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

Martin-Gay, Developmental Mathematics 26 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

Martin-Gay, Developmental Mathematics 27 Input (Animal) Polar Bear Cow Chimpanzee Giraffe Gorilla Kangaroo Red Fox Output (Life Span) Find the domain and range of the following relation. Example Domain and Range

Martin-Gay, Developmental Mathematics 28 Domain is {Polar Bear, Cow, Chimpanzee, Giraffe, Gorilla, Kangaroo, Red Fox} Range is {20, 15, 10, 7} Domain and Range Example continued

Martin-Gay, Developmental Mathematics 29 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 Values  0. Values > 0.

Martin-Gay, Developmental Mathematics 30 Example continued Graphing Piecewise-Defined Functions x y x f (x) = x 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)