Solving Quadratic Equations Chapter Sections 1 Solving Quadratic

Solving Quadratic Equations

Chapter Sections 1. – Solving Quadratic Equations by the Square Root Property 2. – Solving Quadratic Equations by Completing the Square 3. – Solving Quadratic Equations by the Quadratic Formula 4. – Graphing Quadratic Equations in Two Variables 5. – Interval Notation, Finding Domains and Ranges from Graphs and Graphing Piecewise-Defined Functions

Solving Quadratic Equations by the Square Root Property 1.

Square Root Property If b is a real number and a 2 = b, then

Square Root Property Example Solve x 2 = 49 Solve 2 x 2 = 4 x 2 = 2 Solve (y – 3)2 = 4 y=3 2 y = 1 or 5

Square Root Property Example 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 Solve (x + 2)2 = 25 x = 2 ± 5 x = 2 + 5 or x = 2 – 5 x = 3 or x = 7

Square Root Property Example Solve (3 x – 17)2 = 28 3 x – 17 =

Solving Quadratic Equations by Completing the Square 2.

Completing the Square Example What constant term should be added to the following expressions to create a perfect square trinomial? x 2 – 10 x add 52 = 25 x 2 + 16 x add 82 = 64 x 2 – 7 x add

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

Solving Equations Example Solve by completing the square. y 2 + 6 y = 8 y 2 + 6 y + 9 = 8 + 9 (y + 3)2 = 1 y+3=± =± 1 y = 3 ± 1 y = 4 or 2

Solving Equations Example 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 Solve by completing the square. 2 x 2 + 14 x – 1 = 0 2 x 2 + 14 x = 1 x 2 + 7 x = ½ x 2 + 7 x + (x + =½+ )2 = =

3. Solving Quadratic Equations by the Quadratic Formula

The Quadratic Formula A quadratic equation written in standard form, ax 2 + bx + c = 0, has the solutions.

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

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

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

The Discriminant The expression under the radical sign in the formula (b 2 – 4 ac) 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 Example Use the discriminant to determine the number and type of solutions for the following equation. 5 – 4 x + 12 x 2 = 0 a = 12, b = – 4, and c = 5 b 2 – 4 ac = (– 4)2 – 4(12)(5) = 16 – 240 = – 224 There are no real solutions.

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

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

Solving Equations Example Solve the following quadratic equation.

§ 16. 4 Graphing Quadratic Equations in Two Variables

Graphs of Quadratic Equations 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 Example y Graph y = 2 x 2 – 4. x y 2 1 0 – 1 – 2 4 – 2 – 4 – 2 4 (– 2, 4) (2, 4) x (– 1, – 2) (0, – 4)

Intercepts of the Parabola 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.

Characteristics of the Parabola 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. .

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

§ 16. 5 Interval Notation, Finding Domain and Ranges from Graphs, and Graphing Piecewise-Defined Functions

Domain and Range 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 Example 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}

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

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

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

Domain and Range Example continued Domain is {Polar Bear, Cow, Chimpanzee, Giraffe, Gorilla, Kangaroo, Red Fox} Range is {20, 15, 10, 7}

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

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