PLAN
AND
1 PREPARE
Warm-Up Exercises
Transparency Available
Solve the equation.
1. (x 2 5)2 5 49 12, 22
2. (x 1 6)2 5} 20
}
26 1 2Ï5, 26 2 2Ï5
Factor the expression.
3. x 2 1 18x 1 81 (x 1 9)2
4. x 2 2 22x 1 121 (x 2 11)2
5. 27 plus some number is 62.
What is that number? 9
$$!%"
'"
""!%#$ #
( )%'$"'$$
$'$')
'"$ "$#!%"$"
##
)%#&!%$#$
" )$#!%""$#%
#%#$#$#
!%$# "$#!%"$"
Notetaking Guide
&%"$!%$
& Transparency Available
Promotes interactive learning and
notetaking skills, pp. 110–112.
%$
Teaching Resources
For a complete list of lesson
resources, see page 218B.
`#%$#" Pacing
Average: 2 days
Advanced: 2 days
Block: 1 block
• See Teaching Guide/Lesson Plan
in Chapter 5 Resource Book,
pp. 76–77.
FOCUS
$#)%#$$$$( "##
$$ "$#!%"$"# "###
"'$$$"$#"## $$#!%"$#" AND
2 MOTIVATE
x
x
b
x2
bx
Essential Question
How is the process of completing
the square used to solve quadratic
equations? Tell students they
will learn how to answer this
question by learning the process of
completing the square.
x
x2
b2 x
b
b2 x
b2 2
Starting the Lesson
268
b
Ask students what sports they
like that involve hitting a ball. Tell
them that a quadratic equation can
be used to find a ball’s maximum
height if initial height and speed are
known.
x
$" EXAMPLE
2 Complete the Square
3 TEACH
Find the value of c that makes x2 6x c a perfect square trinomial.
Then write the expression as the square of a binomial.
Extra Example 1
Solution
Solve x 2 1 20x 1 100 5 81.
219, 21
To find the value of c, complete the square using b 6.
1 Find
1
2
[(6) 3
half the coefficient of x.
2 Square
3 Replace
Extra Example 2
(3)2 9
the result of Step 1.
c with the result of Step 2.
Find the value of c that makes
x 2 2 26x 1 c a perfect square
trinomial. Then write the expression
as the square of a binomial. 169;
(x 2 13)2
x2 6x 9
ANSWER ` The trinomial x 2 6x c is a perfect square when c 9.
Then x2 6x 9 (x 3)2.
Checkpoint
Key Question to Ask
for Example 2
Perfect Square Trinomials
1. Solve x 2 2x 1 9 by finding square roots. 4, 2
• If x 2 1 bx 1 c 5 0 is a perfect
square trinomial and b is an
odd integer, what do you know
about the value of c? It will be a
fraction.
2. Find the value of c that makes x 2 12x c a perfect square trinomial.
Then write the expression as the square of a binomial. 36; (x 6) 2
Solving Equations Many quadratic equations, such as x2 6x 5 0,
cannot be solved by factoring. You can solve any quadratic equation by
completing the square.
EXAMPLE
Extra Example 3
Solve 2x 2 2 8x 1 12 5 0 by }
completing
the square. 2 1 i Ï2,
}
22i Ï2
3 Solve a Quadratic Equation
Solve 2x2 4x 6 0 by completing the square.
Differentiated Instruction
Solution
STUDENT HELP
AVOID E RRORS
When completing the
square to solve an
equation, you must
always add the same
number to both sides of
the equation.
2x2 4x 6 0
x2 2x 3 0
x2
x2
Below Level Some students have
trouble completing the square
because there are so many steps.
Show them how to break the
process into three parts:
(1) Get the equation into the form
needed for completing the
square.
(2) Complete the square.
(3) Finish the solution by taking
square roots of both sides and
simplifying the results.
If students are making errors,
analyze their work carefully to see
what part of the process is giving
them trouble and give them extra
practice on that part of the process.
Write original equation.
Divide each side by the coefficient of x 2.
Write the left side in the form x 2 bx.
2x 3
2 2
Add [
(1)2 1 to each side.
2
2x 1 3 1
2
(x 1) 2
Write the left side as the square of a binomial.
[
x 1 t2
Take the square root of each side.
[
x 1 t2
Add 1 to each side.
[
x 1 i t2
Write in terms of i.
[
[
ANSWER ` The solutions are 1 i t2 and 1 i t2 .
Checkpoint
Solve a Quadratic Equation by Completing the Square
Solve the equation by completing the square.
[
3. x 2 4x 2 0 2 t6
4. x 2 8x 3 0
[
5. 3w2 6w 12 0 1 i t3
[
4 t13
[
6. w2 12w 4 0 6 4t2
5.8
Completing the Square
269
269
.*,0'*%')-*,!-&,!'&!+ / * !+, .*,0+'%($,!&, +)-*,'/*!,)-*,!-&,!'&
!&.*,0'*%
Extra Example 4
Write y 5 x 2 1 18x 1 95 in vertex
form. Then identify the vertex.
y 5 (x 1 9)2 1 14; vertex (29, 14)
*!, !&.*,0'*% &!&,!1, .*,0
Key Questions to Ask
for Example 4
&% • How do you know what number
to use to complete the square?
You square half of the value of b
in y 5 ax 2 1 bx 1 c.
• Why do you add 25 to each side
in the third step? to complete the
square
"$"!%$
" "$ $$#!%"
$#
"$ # &"
` .*,0'*%!+ .*,0!+
Extra Example 5
The area of the triangle
shown is 144 square units.
What is the value of x ? 8
#%&#%&% #%' #
'-& #1'-*&+/*1*( !&, '*!!&$)-,!'&
2x
$&#%"&% % #
'&,*,'*!+-!$!&#
'&,', +!' '-+ #/!$$
*,&$/!, &*'+)-*, '&,*,'* +,'*!$!&,'-+$'&
+!+', # +!/!$$,$+,
,$'& ,+ '-$, $&, &/!, ', #
&% x 10
Closing the Lesson
Have students summarize the major
points of the lesson and answer
the Essential Question: How is the
process of completing the square
used to solve quadratic equations?
You complete the square so that
one side of the equation can be
written as the square of a binomial.
Then you take square roots of both
sides and simplify the results.
$'$"
#$#$"%$& " "$(
'*
&#(
$#
"$$##$#!%"
$#!%""$#
&"
",, +'$-,!'& -+, +!+', #*,$+,,$'&
` /!, !+, $&, !+,
*!, !&.*,0'*% &!&,!1, .*,0
*,&$ +$&, '&/!, ' *', *,&$!++)-*-&!,+!&, $&, &/!, ', *,&$ 270
!%#