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Example 1 Two Rational RootsExample 2 One Rational RootExample 3 Irrational RootsExample 4 Complex RootsExample 5 Describe Roots
Quadratic Formula Video
• http://www.youtube.com/watch?v=H_7lNT9oDzI
• http://revver.com/video/265387/using-the-quadratic-formula/
Solve by using the Quadratic Formula.
First, write the equation in the form
and identify a, b, and c.
Replace a with 1, b with –8, and c with –33.
Simplify.
Then, substitute these values into the Quadratic Formula.
Quadratic Formula
Answer: The solutions are 11 and –3.
Write as two equations.or
Simplify.
Simplify.
Answer: 2, –15
Solve by using the Quadratic Formula.
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.
Answer: The solution is 17.
Check A graph of the related function shows that there is one solution at
Solve by using the Quadratic Formula.
Answer: 11
Solve by using the Quadratic Formula.
Quadratic Formula
Replace a with 1, b with –6, and c with 2.
Simplify.
Answer: The exact solutions are and The approximate solutions are 0.4 and 5.6.
or
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.
Solve by using the Quadratic Formula.
Answer: or approximately 0.7 and 4.3
Discriminant Video
http://revver.com/video/446510/the-discriminant/
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
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.
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.
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.
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.
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
Assignment
P 318 #14, 18, 20, 24
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
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.
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)
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.
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
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.
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.
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).
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.
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.
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.
Subtract 2 from each side.
Divide each side by 4.
Answer: The equation of the parabola in vertex form is
Check A graph of verifies that the
parabola passes through the point at (3, 4).
Answer:
Write an equation for the parabola whose vertex is at
(2, 3) and passes through (–2, 1).
Assignment
• P 326 #16, 20, 30, 32
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
Graphing Quadratic Inequalities
http://www.mathwarehouse.com/quadratic-inequality/how-to-solve-and-graph-quadratic-inequality.php
Graph
Step 1 Graph the related quadraticequation, Since the inequality symbolis >, the parabola should bedashed.
Graph
Step 2Test a point inside the parabola, such as
(1, 2).
So, (1, 2) is a solution of the inequality.
(1, 2)
(1, 2)(1, 2)
Step 3 Shade the region inside the parabola.
Graph
Answer:
Graph
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
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
Solve by graphing.
Answer:
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.
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.
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:
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.
Solve by by graphing.
Answer:
Solve algebraically.
First, solve the related equation .
Related quadratic equation
Subtract 2 from each side.
Factor.
Solve each equation.
Zero Product Propertyor
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.
Test a value in each interval to see if it satisfies the original inequality.
Answer: The solution set is This is shown on the number line below.
Solve algebraically.
Answer:
Assignment
P 333 #14, 18, 26, 28
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