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Lesson 5 Contents Example 1Two Rational Roots Example 2One Rational Root Example 3Irrational Roots Example 4Complex Roots Example 5Describe Roots

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Page 1: Lesson 5 Contents Example 1Two Rational Roots Example 2One Rational Root Example 3Irrational Roots Example 4Complex Roots Example 5Describe Roots
Page 2: Lesson 5 Contents Example 1Two Rational Roots Example 2One Rational Root Example 3Irrational Roots Example 4Complex Roots Example 5Describe Roots

Example 1 Two Rational RootsExample 2 One Rational RootExample 3 Irrational RootsExample 4 Complex RootsExample 5 Describe Roots

Page 3: Lesson 5 Contents Example 1Two Rational Roots Example 2One Rational Root Example 3Irrational Roots Example 4Complex Roots Example 5Describe Roots

Quadratic Formula Video

• http://www.youtube.com/watch?v=H_7lNT9oDzI

• http://revver.com/video/265387/using-the-quadratic-formula/

http://www.youtube.com/watch?v=H_7lNT9oDzI
http://www.youtube.com/watch?v=H_7lNT9oDzI
http://revver.com/video/265387/using-the-quadratic-formula/
http://revver.com/video/265387/using-the-quadratic-formula/
Page 4: Lesson 5 Contents Example 1Two Rational Roots Example 2One Rational Root Example 3Irrational Roots Example 4Complex Roots Example 5Describe Roots

Solve by using the Quadratic Formula.

First, write the equation in the form

and identify a, b, and c.

Page 5: Lesson 5 Contents Example 1Two Rational Roots Example 2One Rational Root Example 3Irrational Roots Example 4Complex Roots Example 5Describe Roots

Replace a with 1, b with –8, and c with –33.

Simplify.

Then, substitute these values into the Quadratic Formula.

Quadratic Formula

Page 6: Lesson 5 Contents Example 1Two Rational Roots Example 2One Rational Root Example 3Irrational Roots Example 4Complex Roots Example 5Describe Roots

Answer: The solutions are 11 and –3.

Write as two equations.or

Simplify.

Simplify.

Page 7: Lesson 5 Contents Example 1Two Rational Roots Example 2One Rational Root Example 3Irrational Roots Example 4Complex Roots Example 5Describe Roots

Answer: 2, –15

Solve by using the Quadratic Formula.

Page 8: Lesson 5 Contents Example 1Two Rational Roots Example 2One Rational Root Example 3Irrational Roots Example 4Complex Roots Example 5Describe Roots

Replace a with 1, b with –34, and c with 289.

Identify a, b, and c. Then, substitute these values into the Quadratic Formula.

Solve by using the Quadratic Formula.

Quadratic Formula

Simplify.

Page 9: Lesson 5 Contents Example 1Two Rational Roots Example 2One Rational Root Example 3Irrational Roots Example 4Complex Roots Example 5Describe Roots

Answer: The solution is 17.

Check A graph of the related function shows that there is one solution at

Page 10: Lesson 5 Contents Example 1Two Rational Roots Example 2One Rational Root Example 3Irrational Roots Example 4Complex Roots Example 5Describe Roots

Solve by using the Quadratic Formula.

Answer: 11

Page 11: Lesson 5 Contents Example 1Two Rational Roots Example 2One Rational Root Example 3Irrational Roots Example 4Complex Roots Example 5Describe Roots

Solve by using the Quadratic Formula.

Quadratic Formula

Replace a with 1, b with –6, and c with 2.

Simplify.

Page 12: Lesson 5 Contents Example 1Two Rational Roots Example 2One Rational Root Example 3Irrational Roots Example 4Complex Roots Example 5Describe Roots

Answer: The exact solutions are and The approximate solutions are 0.4 and 5.6.

or

Page 13: Lesson 5 Contents Example 1Two Rational Roots Example 2One Rational Root Example 3Irrational Roots Example 4Complex Roots Example 5Describe Roots

Check Check these results by graphing the related quadratic function,

Using the ZERO function of a graphing calculator, the approximate zeros of the related function are

–2.9 and 0.9.

Page 14: Lesson 5 Contents Example 1Two Rational Roots Example 2One Rational Root Example 3Irrational Roots Example 4Complex Roots Example 5Describe Roots

Solve by using the Quadratic Formula.

Answer: or approximately 0.7 and 4.3

Page 15: Lesson 5 Contents Example 1Two Rational Roots Example 2One Rational Root Example 3Irrational Roots Example 4Complex Roots Example 5Describe Roots

Discriminant Video

http://revver.com/video/446510/the-discriminant/

Page 16: Lesson 5 Contents Example 1Two Rational Roots Example 2One Rational Root Example 3Irrational Roots Example 4Complex Roots Example 5Describe Roots

Discriminant and Types of Roots

Value of Discriminant Type and # of Roots Example Graph

b2 – 4ac > 0 and is a 2 real, rational rootsperfect square

b2 – 4ac > 0 and is not a 2 real, irrational rootsperfect square

b2 – 4ac = 0 1 real, rational root

b2 – 4ac < 0 2 complex roots

Page 17: Lesson 5 Contents Example 1Two Rational Roots Example 2One Rational Root Example 3Irrational Roots Example 4Complex Roots Example 5Describe Roots

Answer: The discriminant is 0, so there is one rational root.

Find the value of the discriminant for . Then describe the number and type of roots for the equation.

Page 18: Lesson 5 Contents Example 1Two Rational Roots Example 2One Rational Root Example 3Irrational Roots Example 4Complex Roots Example 5Describe Roots

Answer: The discriminant is negative, so there are two complex roots.

Find the value of the discriminant for . Then describe the number and type of roots for the equation.

Page 19: Lesson 5 Contents Example 1Two Rational Roots Example 2One Rational Root Example 3Irrational Roots Example 4Complex Roots Example 5Describe Roots

Answer: The discriminant is 80, which is not a perfect square. Therefore, there are two irrational roots.

Find the value of the discriminant for . Then describe the number and type of roots for the equation.

Page 20: Lesson 5 Contents Example 1Two Rational Roots Example 2One Rational Root Example 3Irrational Roots Example 4Complex Roots Example 5Describe Roots

Answer: The discriminant is 81, which is a perfect square. Therefore, there are two rational roots.

Find the value of the discriminant for . Then describe the number and type of roots for the equation.

Page 21: Lesson 5 Contents Example 1Two Rational Roots Example 2One Rational Root Example 3Irrational Roots Example 4Complex Roots Example 5Describe Roots

Find the value of the discriminant for each quadratic equation. Then describe the number and type of roots for the equation.

a.

b.

c.

d.

Answer: 0; 1 rational root

Answer: –24; 2 complex roots

Answer: 5; 2 irrational roots

Answer: 64; 2 rational roots

Page 22: Lesson 5 Contents Example 1Two Rational Roots Example 2One Rational Root Example 3Irrational Roots Example 4Complex Roots Example 5Describe Roots
Page 23: Lesson 5 Contents Example 1Two Rational Roots Example 2One Rational Root Example 3Irrational Roots Example 4Complex Roots Example 5Describe Roots

Assignment

P 318 #14, 18, 20, 24

Page 24: Lesson 5 Contents Example 1Two Rational Roots Example 2One Rational Root Example 3Irrational Roots Example 4Complex Roots Example 5Describe Roots

Example 1 Graph a Quadratic Function in Vertex Form

Example 2 Write y = x2 + bx + c in Vertex Form

Example 3 Write y = ax2 + bx + c in Vertex Form, a 1Example 4 Write an Equation Given Points

Page 25: Lesson 5 Contents Example 1Two Rational Roots Example 2One Rational Root Example 3Irrational Roots Example 4Complex Roots Example 5Describe Roots

Now use this information to draw the graph.

Analyze Then draw its graph.

h = 3 and k = 2

Answer: The vertex is at (h, k) or (3, 2) and the axis of symmetry is The graph has the same shape as the graph of but is translated 3 units right and 2 units up.

Page 26: Lesson 5 Contents Example 1Two Rational Roots Example 2One Rational Root Example 3Irrational Roots Example 4Complex Roots Example 5Describe Roots

Step 1

Plot the vertex, (3, 2).

Step 3Find and plot two points on one side of the axis of symmetry, such as

(2, 3) and (1, 6).

Step 4Use symmetry to complete the graph.

Step 2Draw the axis of symmetry,

(1, 6) (5, 6)

(2, 3) (4, 3)

(3, 2)

Page 27: Lesson 5 Contents Example 1Two Rational Roots Example 2One Rational Root Example 3Irrational Roots Example 4Complex Roots Example 5Describe Roots

Answer:

Analyze Then draw its graph.

The vertex is at (–2, –4), and the axis of symmetry isThe graph has the same shape as the graph of

; it is translated 2 units left and 4 units down.

Page 28: Lesson 5 Contents Example 1Two Rational Roots Example 2One Rational Root Example 3Irrational Roots Example 4Complex Roots Example 5Describe Roots

Write in vertex form. Then analyze the function.

Notice that is not a perfect square.

Balance this addition

by subtracting 1.

Complete the square by

adding

Page 29: Lesson 5 Contents Example 1Two Rational Roots Example 2One Rational Root Example 3Irrational Roots Example 4Complex Roots Example 5Describe Roots

This function can be rewritten asSo, and

Answer: The vertex is at (–1, 3), and the axis of symmetry is Since the graph opens up and has the

same shape as but is translated 1 unit left and 3 units up.

Writeas a perfect square.

Page 30: Lesson 5 Contents Example 1Two Rational Roots Example 2One Rational Root Example 3Irrational Roots Example 4Complex Roots Example 5Describe Roots

Write in vertex form. Then analyze the function.

Answer: vertex: (–3, –4); axis of symmetry:

opens up; the graph has the same shape as the graph of but it is

translated 3 units left and 4 units down.

Page 31: Lesson 5 Contents Example 1Two Rational Roots Example 2One Rational Root Example 3Irrational Roots Example 4Complex Roots Example 5Describe Roots

Write in vertex form. Then analyze and graph the function.

Original equation

Group and

factor, dividing by a.

Complete the square by adding

1 inside the parentheses.

Balance this addition by

subtracting –2(1).

Page 32: Lesson 5 Contents Example 1Two Rational Roots Example 2One Rational Root Example 3Irrational Roots Example 4Complex Roots Example 5Describe Roots

Write as a perfect square.

Now graph the function. Two points on the graph to the right of are (0, 2) and (0.5, –0.5). Use symmetry to complete the graph.

Answer: The vertex form is So,

and The vertex is at (–1, 4) and the axis of symmetry is Since the graph opens down and is

narrower than It is also translated 1 unit left and 4 units up.

Page 33: Lesson 5 Contents Example 1Two Rational Roots Example 2One Rational Root Example 3Irrational Roots Example 4Complex Roots Example 5Describe Roots
Page 34: Lesson 5 Contents Example 1Two Rational Roots Example 2One Rational Root Example 3Irrational Roots Example 4Complex Roots Example 5Describe Roots

Write in vertex form. Then analyze and graph the function.

Answer:

vertex: (–1, 7); axis of symmetry:

x = –1; opens down; the graph is narrower than the graph of y = x2,and it is

translated 1 unit left and 7 units up.

Page 35: Lesson 5 Contents Example 1Two Rational Roots Example 2One Rational Root Example 3Irrational Roots Example 4Complex Roots Example 5Describe Roots

Write an equation for the parabola whose vertex is at

(1, 2) and passes through (3, 4).

The vertex of the parabola is at (1, 2) so and Since (3, 4) is a point on the graph of the parabola , and Substitute these values into the vertex form of the equation and solve for a.

Vertex form

Substitute 1 for h, 2 for k, 3 for x, and 4 for y.

Simplify.

Page 36: Lesson 5 Contents Example 1Two Rational Roots Example 2One Rational Root Example 3Irrational Roots Example 4Complex Roots Example 5Describe Roots

Subtract 2 from each side.

Divide each side by 4.

Answer: The equation of the parabola in vertex form is

Page 37: Lesson 5 Contents Example 1Two Rational Roots Example 2One Rational Root Example 3Irrational Roots Example 4Complex Roots Example 5Describe Roots

Check A graph of verifies that the

parabola passes through the point at (3, 4).

Page 38: Lesson 5 Contents Example 1Two Rational Roots Example 2One Rational Root Example 3Irrational Roots Example 4Complex Roots Example 5Describe Roots

Answer:

Write an equation for the parabola whose vertex is at

(2, 3) and passes through (–2, 1).

Page 39: Lesson 5 Contents Example 1Two Rational Roots Example 2One Rational Root Example 3Irrational Roots Example 4Complex Roots Example 5Describe Roots
Page 40: Lesson 5 Contents Example 1Two Rational Roots Example 2One Rational Root Example 3Irrational Roots Example 4Complex Roots Example 5Describe Roots

Assignment

• P 326 #16, 20, 30, 32

Page 41: Lesson 5 Contents Example 1Two Rational Roots Example 2One Rational Root Example 3Irrational Roots Example 4Complex Roots Example 5Describe Roots

Example 1 Graph a Quadratic Inequality

Example 2 Solve ax2 + bx + c 0

Example 3 Solve ax2 + bx + c 0Example 4 Write an InequalityExample 5 Solve a Quadratic Inequality

Page 42: Lesson 5 Contents Example 1Two Rational Roots Example 2One Rational Root Example 3Irrational Roots Example 4Complex Roots Example 5Describe Roots

Graphing Quadratic Inequalities

http://www.mathwarehouse.com/quadratic-inequality/how-to-solve-and-graph-quadratic-inequality.php

http://www.mathwarehouse.com/quadratic-inequality/how-to-solve-and-graph-quadratic-inequality.php
http://www.mathwarehouse.com/quadratic-inequality/how-to-solve-and-graph-quadratic-inequality.php
http://www.mathwarehouse.com/quadratic-inequality/how-to-solve-and-graph-quadratic-inequality.php
Page 43: Lesson 5 Contents Example 1Two Rational Roots Example 2One Rational Root Example 3Irrational Roots Example 4Complex Roots Example 5Describe Roots

Graph

Step 1 Graph the related quadraticequation, Since the inequality symbolis >, the parabola should bedashed.

Page 44: Lesson 5 Contents Example 1Two Rational Roots Example 2One Rational Root Example 3Irrational Roots Example 4Complex Roots Example 5Describe Roots

Graph

Step 2Test a point inside the parabola, such as

(1, 2).

So, (1, 2) is a solution of the inequality.

(1, 2)

Page 45: Lesson 5 Contents Example 1Two Rational Roots Example 2One Rational Root Example 3Irrational Roots Example 4Complex Roots Example 5Describe Roots

(1, 2)(1, 2)

Step 3 Shade the region inside the parabola.

Graph

Page 46: Lesson 5 Contents Example 1Two Rational Roots Example 2One Rational Root Example 3Irrational Roots Example 4Complex Roots Example 5Describe Roots

Answer:

Graph

Page 47: Lesson 5 Contents Example 1Two Rational Roots Example 2One Rational Root Example 3Irrational Roots Example 4Complex Roots Example 5Describe Roots

The solution consists of the x values for which the graph of the related quadratic function lies above the x-axis. Begin by finding the roots of the related equation.

Solve by graphing.

Related equation

Factor.

Solve each equation.

Zero Product Propertyor

Page 48: Lesson 5 Contents Example 1Two Rational Roots Example 2One Rational Root Example 3Irrational Roots Example 4Complex Roots Example 5Describe Roots

Sketch the graph of the parabola that has x-intercepts at 3 and 1. The graph lies above the x-axis to the left of

and to the right of

Answer: The solution set is

Page 49: Lesson 5 Contents Example 1Two Rational Roots Example 2One Rational Root Example 3Irrational Roots Example 4Complex Roots Example 5Describe Roots

Solve by graphing.

Answer:

Page 50: Lesson 5 Contents Example 1Two Rational Roots Example 2One Rational Root Example 3Irrational Roots Example 4Complex Roots Example 5Describe Roots

Solve by graphing.

This inequality can be rewritten as The solution consists of the x-values for which the graph of the related quadratic equation lies on and above the x-axis. Begin by finding roots of the related equation.

Related equation

Use the Quadratic Formula.

Replace a with –2,

b with –6 and c with 1.

Page 51: Lesson 5 Contents Example 1Two Rational Roots Example 2One Rational Root Example 3Irrational Roots Example 4Complex Roots Example 5Describe Roots

Simplify and write as two equations.

or

Simplify.

Sketch the graph of the parabola that has x-intercepts of –3.16 and 0.16. The graph should open down since a < 0.

Page 52: Lesson 5 Contents Example 1Two Rational Roots Example 2One Rational Root Example 3Irrational Roots Example 4Complex Roots Example 5Describe Roots

The graph lies on and above the x-axis at and and between these two values. The solution set

of the inequality is approximately

Answer:

Page 53: Lesson 5 Contents Example 1Two Rational Roots Example 2One Rational Root Example 3Irrational Roots Example 4Complex Roots Example 5Describe Roots

Check Test one value of x less than –3.16, one between –3.16 and 0.16, and one greater than 0.16 in the original inequality.

Page 54: Lesson 5 Contents Example 1Two Rational Roots Example 2One Rational Root Example 3Irrational Roots Example 4Complex Roots Example 5Describe Roots

Solve by by graphing.

Answer:

Page 55: Lesson 5 Contents Example 1Two Rational Roots Example 2One Rational Root Example 3Irrational Roots Example 4Complex Roots Example 5Describe Roots

Solve algebraically.

First, solve the related equation .

Related quadratic equation

Subtract 2 from each side.

Factor.

Solve each equation.

Zero Product Propertyor

Page 56: Lesson 5 Contents Example 1Two Rational Roots Example 2One Rational Root Example 3Irrational Roots Example 4Complex Roots Example 5Describe Roots

Plot –2 and 1 on a number line. Use closed circles since these solutions are included. Notice that the number line is separated into 3 intervals.

Page 57: Lesson 5 Contents Example 1Two Rational Roots Example 2One Rational Root Example 3Irrational Roots Example 4Complex Roots Example 5Describe Roots

Test a value in each interval to see if it satisfies the original inequality.

Page 58: Lesson 5 Contents Example 1Two Rational Roots Example 2One Rational Root Example 3Irrational Roots Example 4Complex Roots Example 5Describe Roots

Answer: The solution set is This is shown on the number line below.

Page 59: Lesson 5 Contents Example 1Two Rational Roots Example 2One Rational Root Example 3Irrational Roots Example 4Complex Roots Example 5Describe Roots

Solve algebraically.

Answer:

Page 60: Lesson 5 Contents Example 1Two Rational Roots Example 2One Rational Root Example 3Irrational Roots Example 4Complex Roots Example 5Describe Roots

Assignment

P 333 #14, 18, 26, 28

Page 61: Lesson 5 Contents Example 1Two Rational Roots Example 2One Rational Root Example 3Irrational Roots Example 4Complex Roots Example 5Describe Roots

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