1.2 Quadratic Equations

1
Chapter 1 Section 2 Quadratic Equations

• In this section, we will
• Solve quadratic equations by factoring
• Solve quadratic equations by extraction of roots
• Solve quadratic equations by completing the
  square
• Solve quadratic equations by using the quadratic
  formula
• (real solutions only)
• Use the discriminant to determine the nature of
  the
• real solutions of a given quadratic equation
• Solve applications involving quadratic equations

2
quadratic equation
standard form

• We will solve quadratic equations by
• factoring
• extraction of roots
• completing the square
• using the quadratic formula
• possible outcomes

1.2 Quadratic Equations
3

• Solving Quadratic Equations by Factoring
• 1. Write the quadratic equation in standard form
• (i.e. set the equation equal to 0)
• 2. Factor completely
• 3. Use the Zero-Product Rule
• 4. Solve the resulting linear equations
• 5. Check your potential solution(s)

• We will solve by
• factoring when,
• once set equal to
• zero, the result
• is factorable.

1.2 Quadratic Equations Solving Quadratic
Equations by Factoring
4
Examples Solve each equation by factoring.
Check your result(s).
checks
check
1.2 Quadratic Equations Solving Quadratic
Equations by Facoring
5
Example Solve the equation by factoring. Check
your result(s).
checks
1.2 Quadratic Equations Solving Quadratic
Equations by Factoring
6

• Solving Quadratic Equations by Extracting Roots
• 1. Write the quadratic equation in the form
• 2. Take the square root of both sides of the
  equation
• If c lt 0, there will be no real solutions
• If c 0, there will be one real solution
• If c gt 0, there will be two real solutions
• 3. Check your potential solution(s)

• We will solve by
• extracting roots
• when our equation
• has the form

1.2 Quadratic Equations Solving Quadratic
Equations by Extraction of Roots
7
Examples Solve each equation by extracting the
roots (a.k.a. the square root method). Check
your result(s).
checks
checks
1.2 Quadratic Equations Solving Quadratic
Equations by Extraction of Roots
8
Completing the Square Recall our perfect squares
from Review Section 4 example We will be
reversing this process and filling in the
blanks. examples
Special Product Formulas Special Product Formulas
Perfect Squares
Perfect Squares
Take half of the x-term coefficient
It goes here
Now square that result
It goes here
1.2 Quadratic Equations Solving Quadratic
Equations by Completing the Square
9

• Solving Quadratic Equations by Completing the
  Square
• 1. Write the quadratic equation in the form
• 2. Make sure a 1
• if it is not, divide all terms by a
• 3. Complete the square
• Find half of b
• Square the result
• 4. Solve the resulting equation by extraction of
  roots
• 5. Check your potential solution(s)

• We can solve
• any quadratic eq.
• this way!

1.2 Quadratic Equations Solving Quadratic
Equations by Completing the Square
10
Examples Solve the equation by completing the
square. Check your result(s).
checks
1.2 Quadratic Equations Solving Quadratic
Equations by Completing the Square
11
Examples Solve the equation by completing the
square. Check your result(s).
checks
1.2 Quadratic Equations Solving Quadratic
Equations by Completing the Square
12
Examples Solve the equation by completing the
square. Check your result(s).
checks
1.2 Quadratic Equations Solving Quadratic
Equations by Completing the Square
13
Examples Solve the equation by completing the
square. Check your result(s).
checks
1.2 Quadratic Equations Solving Quadratic
Equations by Completing the Square
14
Solving Quadratic Equations by Using the
Quadratic Formula 1. Write the quadratic
equation in the form 2. Use the quadratic
formula to find the solution(s) 3. Check your
potential solution(s)

• We can solve
• any quadratic eq.
• this way!

• We can use the discriminant to determine the
  nature of the real solutions to our given
  quadratic equation.
• If the discriminant of is
• negative, then there are no real solutions
• zero, then there is one real solution
• positive, then there are two different real
  solutions

1.2 Quadratic Equations Solving Quad. Equations
by Using the Quadratic Formula
15
Examples Solve the equation by using the
quadratic formula Check your result(s).
1.2 Quadratic Equations Solving Quad. Equations
by Using the Quadratic Formula
16
Examples Solve the equation by using the
quadratic formula Check your result(s).
1.2 Quadratic Equations Solving Quad. Equations
by Using the Quadratic Formula
17
Examples Solve the equation by using the
quadratic formula Check your result(s).
1.2 Quadratic Equations Solving Quad. Equations
by Using the Quadratic Formula
18
Examples Use the discriminant to determine the
nature of the real solutions of the following
quadratic equations.
1.2 Quadratic Equations Using the Discriminant
to Determine the Nature of Solutions
19

• Summary of Techniques
• We can now solve quadratic equations by
• Factoring
• place in form
• factor and use zero-product rule
• Extracting the roots
• place in form
• take the square root of both sides
• Completing the square
• place in form
• complete the square
• take the square root of both sides
• Using the quadratic formula
• place in the form
• use the quadratic formula

Can only solve by factoring if this is factorable
Can only extract roots if there is no x-term
Take half of the x-term coefficient
It goes here
Can use to solve any quad. eq.
Now square that result
It goes here
1.2 Quadratic Equations
20

• How to Solve a Word Problem
• Step 1 Read the problem until you understand it.
  
• What are we asked to find?
• What information is given?
• What vocabulary is being used?
• Step 2 Assign a variable to represent what you
  are looking for.
• Express any remaining unknown
  quantities in terms of this variable.
• Step 3 Make a list of all known facts and form
  an equation or inequality to solve.
• It may help to make a labeled
  diagram, table or chart, graph
• Step 4 Solve
• Step 5 State the solution in a complete sentence
  by mirroring the original question.
• Be sure to include units when
  necessary.
• Step 6 Check your result(s) in the words of the
  problem
• Does your solution make sense?

1.2 Quadratic Equations Solving Applications
Involving Quadratic Equations
21
Example The median weekly earnings E, in
dollars, for full-time women workers ages 16
years and older from 2000 through 2008 can be
estimated by the equation
where x is the number of years
after 2000. In what year will the median weekly
earnings be 632.
1.2 Quadratic Equations Solving Applications
Involving Quadratic Equations
22
Example The area of a rectangular window is to
be 143 square feet. If the length is to be two
feet more than the width, what are the dimensions?
1.2 Quadratic Equations Solving Applications
Involving Quadratic Equations
23

• Example A ball is thrown upward with an initial
  velocity of 20 meters per second. The distance
  s, in meters, of the object from the ground after
  t seconds is
• When will the object be 15 meters above the
  ground?
• When will it strike the ground?

1.2 Quadratic Equations Solving Applications
Involving Quadratic Equations
24
Independent Practice You learn math by doing
math. The best way to learn math is to practice,
practice, practice. The assigned homework
examples provide you with an opportunity to
practice. Be sure to complete every assigned
problem (or more if you need additional
practice). Check your answers to the
odd-numbered problems in the back of the text to
see whether you have correctly solved each
problem rework all problems that are
incorrect. Read pp. 97-106 Homework pp.
106-109 15-23 odds, 29-37 odds, 41-45 odds,
53-59 odds, 71, 93-101 odds, 107
1.2 Quadratic Equations