Quadratic
Equations
Math 20 – Pre-Calculus
Chapter 4
Name:__________________
Class:___________________
1
Relations and Functions
Specific Outcomes
5. Solve problems that involve
quadratic equations.
General Outcome:
Develop algebraic and graphical reasoning through the study of
relations
Achievement Indicators:
The following set of indicators may be used to determine whether
students have met the corresponding specific outcome
5.1
Explain, using examples, the relationship among the roots
of a quadratic equation, the zeros of the corresponding
quadratic function and the x-intercepts of the graph of the
quadratic function.
5.2
Derive the quadratic formula, using deductive reasoning.
5.3
5.4
5.5
5.6
5.7
Solve a quadratic equation of the form ax  bx  c  0 by
using strategies such as:
 Determining the square roots
 Factoring
 Completing the square
 Applying the quadratic formula
 Graphing its corresponding function
Select a method for solving a quadratic equation, justify
the choice, and verify the solution.
Explain, using examples, how the discriminant may be
used to determine whether a quadratic equation has two,
one or no real roots; and relate the number of zeros to the
graph of the corresponding quadratic function.
Identify and correct errors in a solution to a quadratic
equation.
Solve a problem by:
 Analyzing a quadratic equation
 Determining and analyzing a quadratic equation.
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Big Ideas:
Students will understand …

Information from quadratic equations and the graphs of quadratic
functions can be used to solve problems in areas like physics, calculus,
engineering, architecture, sports and business.
By the end of the unit students should:



The basic shape and characteristics of the graph of a quadratic function.
Quadratic functions and equations relate to real life situations.
Solving an equation means to find the root(s).
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4.1 Graphical Solutions of
Quadratic Equations
Part A: Graphically Solutions
Recall from Chapter 3
A quadratic function is …
A quadratic equation is …
Root(s) of an equation…
Zeros(s) of a function …
The strategies that we will learn to solve a quadratic equation are:
- Graphing it’s corresponding function
- Factoring
- Completing the square
- The Quadratic Formula
Example: Graph each of quadratic function and determine the x-intercepts
y  x2  8x  5
y  2 x 2  12 x  10
3
Example: Solve each of the quadratic equation by using a graphical approach
a) 4 x 2  12 x  1  0
b) x 2  x  0
Summary
Types of Quadratic Functions
4
c) 3x 2  10 x  9  0
Part B: Applying Graphically Solutions to Problems
How do I approach a “word” problem?
Example: The height, h, in meters above the ground, of a ball being thrown off a roof at
time, t after the launch is defined by the function h(t )  4t 2  48t  3 .
a) Sketch the important part of the parabola.
b) Find the maximum height reached by the ball. How long does it take the ball to reach
it’s maximum height?
c) How long does it take for the ball to hit the ground?
d) How high is the roof?
e) Find the height of the ball after 3 seconds.
Homework
Textbook Page 215 #1 - 6, 9, 10
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4.2 Factoring Quadratic Equations
Part I: Review of Factoring
Recall from Math 10 – C
Key Ideas regarding factoring
Example: Factor each of the following
a. y 2  8 y  15
b. 12 x 2  2 x  30
c. 4t 2  16t  48
d. 4 x 2  12 x  9
e. 25m2  36
f. 8x  50 xy 2
g. ( x  5)2  4( x  5)  21
3
h. 4( y  2)2  36( x  3) 2
Homework
Textbook Page 229 #1-6
6
4.2 Factoring Quadratic Equations
Part II: Solving Quadratic Equations by Factoring
Recall: If (A)(B) = 0, then either A = 0 or B = 0
What is the difference between factoring and solving an equation?
Example: Solve each of the following quadratic equations by factoring.
a. x 2  3x  2  0
2
b. x  8x  15  0
c. 3x 2  14 x  5  0
2
d. 2 x  24 x  54  0
e. x 2  64  0
f. x 2  10 x  16
g. x 2  9 x  0
h. 25 x 2  49  0
i.
Homework
Textbook Page 230 #7-10, 12, 13
7
1 2
x  5 x  18
3
4.3 Solving Quadratic Equations
by Completing the Square
Recall from Chapter 3
Recall: Perfect Square Trinomials
(a  b)2  a 2  2ab  b2
Indicate the constant that must be added so that the trinomial is a perfect square and then
factor the trinomial.
a. x2 + 14x +
b. x2 – 20x +
Convert each of the following functions into vertex form
a. y  x 2  18 x  32
b. y  x 2  8t  4
Example: Solve each quadratic equation by completing the square.
a. x 2  10 x  23  0
b. a 2  6a  7  0
c. 2 x 2  12 x  11  0
Homework
Textbook Page 240 #1, 3, 6
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4.4 The Quadratic Formula
Part I: The Quadratic Formula
Complete the square to solve this quadratic equation: (b and c are constants)
ax 2  bx  c  0
The roots(solutions) of the quadratic equation ax 2  bx  c  0 can be found using the
Quadratic Formula:
x
b  b 2  4ac
2a
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Example: Use the quadratic formula to solve each of the following quadratic equations.
a. 3x 2  5x  11  0
a=
b=
b. 2t 2  11t  13  0
c. x 2  6x  7  0
c=
Example: The height of the Peace Tower in Ottawa is 90 m. If an object is thrown
downward at 5 m/s from this height, the time, t seconds, the object takes to
reach the ground can be found by solving the equation –4.9t2-5t + 90 = 0.
Find the time, to the nearest tenth of a second, for the object to hit the ground?
Homework
Textbook Page 254 #3, 4, 5
10
4.4 The Quadratic Formula
Part II: The Discriminant
Graph each of the following quadratic equations and use the quadratic formula to
determine the roots of each function.
x2  2x  3  0
x2  2 x  1  0
x2  2x  3  0
If you examine the expression, b 2  4ac , in the quadratic formula the result will help us
determine the roots. b 2  4ac is called the discriminant of the quadratic equation because
it discriminates among the types of possible solutions.
Number of Roots of a Quadratic Equation:
The quadratic equation ax 2  bx  c  0 has:

two real roots when
b 2  4ac  0

exactly one real root when
b 2  4ac  0

no real roots when
b 2  4ac  0
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Example: Use the discriminant to determine the number of roots for each. Use
technology to check your answers graphically.
a. 16 x 2  8 x  1  0
b. 3 x 2  6 x  3  0
c. 2 x 2  7 x  9  0
Nature of the Roots
Example
Determine the value of “m” that will give the quadratic:
3x 2  4 x  m  0
a. two equal roots (1 real root)
b. no real roots
mx 2  8 x  3  0
a. two distinct real roots
Homework
Textbook Page 254 #1, 2
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