Chapter 9 Review Square Roots and Radicals Try some: Simplify these:
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Chapter 9 ReviewChapter 9 Review
Square Roots and RadicalsSquare Roots and Radicals
• Try some:
• Simplify these: 25
1
0
144
16€
±5
€
±1
€
0
€
±12
€
No Solution
• More examples:144
48
196€
±12
€
±4 3
€
±13
How would you solve the equation:How would you solve the equation:xx22 = 4 = 4
(take the square root of each side!)(take the square root of each side!)
* Remember, the square root of a positive # has 2 answers! (one + and one -)
*2- 2
4
4
2
2
orx
x
x
Solving Quadratic EquationsSolving Quadratic Equations1. Solve. 3 - 5x2 = -9
-3 -3
-5x2 = -12
-5 -5
x2 = 5
12
5
122 x
5
152
5
15*4
25
60
5*5
5*12
5
12 x
2.Solve. 3(x-2)2=21 3 3
(x-2)2 = 7
7)2( 2 x
72 x
72x
More Examples!More Examples!
3. Solve. 4x2-6=42
+6 +6
4x2=48
4 4
x2 = 12
122 x
323*4 x
4. Solve. 4. Solve. 6)4(5
1 2 x
30)4( 2 x
30)4( 2 x
304 x
304x
Rationalizing the DenominatorRationalizing the Denominator
You CANNOT leave a radical in the denominator of a fraction!
No tents in the basement!!!!
(the numerator is OK)
Just multiply the top & bottom of the fraction by the radical to “rationalize” the
denominator.
Properties of Square RootsProperties of Square Roots
(a>0 and b>0)
1. Product Property –
2. Quotient Property-
baab *
Example:Example:
10210*410*440
b
a
b
a
Example:Example:
2
3
4
3
4
3
29
29
18
23
96
616
616
64
12 45
34
34
32
59
59
53
€
5( )2
55
55
25
5
€
7( )2
77
77
49
7
€
10( )2
1010
1010
100
10
2473
2743
2712
1412
3536 3356
3330
930
330
90
10528
10258
10240 2040
5440
5240
580
25147 21457
21435 2835
7435
7235
770
Find a perfect square factor of 32.Find a perfect square factor of 32.
Simplify each expression. Simplify each expression.
Product Property of Square RootsProduct Property of Square Roots
Quotient Property of Square RootsQuotient Property of Square Roots
A.A.
B.B.
Product Property of Square RootsProduct Property of Square Roots
Simplify each expression. Simplify each expression.
Quotient Property of Square RootsQuotient Property of Square Roots
CC..
D.D.
E.E.
Simplify each expression.Simplify each expression.
F.F.
Find a perfect square factor of 48.Find a perfect square factor of 48.
Product Property of Square RootsProduct Property of Square Roots
Quotient Property of Square RootsQuotient Property of Square Roots
Simplify.Simplify.
Simplify each expression.Simplify each expression.
G.G.
H.H.
Product Property of Square RootsProduct Property of Square Roots
Quotient Property of Square RootsQuotient Property of Square Roots
ExamplesExamples
1.
2.
3.
500 5*100 5*100 510
6*123 6*123 723 2*363
26*3 218
9
25
9
25
3
5
More Examples!More Examples!
1.
2.
3
25
3
25
3
5
3*
3*
9
35
3
35
Can’t have a tent in the Can’t have a tent in the basement!basement!
11
2
11
2
11*
11*
121
22
11
22
Simplify by rationalizing the denominator. Simplify by rationalizing the denominator.
Multiply by a form of 1.Multiply by a form of 1.
= 2= 2
Simplify the expression. Simplify the expression.
Multiply by a form of 1.Multiply by a form of 1.
Simplify by rationalizing the denominator.Simplify by rationalizing the denominator.
Multiply by a form of 1.Multiply by a form of 1.
Simplify by rationalizing the denominator.Simplify by rationalizing the denominator.
Multiply by a form of 1.Multiply by a form of 1.
1.1. Estimate to the nearest tenth. Estimate to the nearest tenth. 6.76.7Simplify each expression.Simplify each expression.
2. 2.
3.3.
4.4.
5.5.
Solving Quadratic Equations using the Quadratic Solving Quadratic Equations using the Quadratic FormulaFormula
General equation of a General equation of a quadratic:quadratic:
02 cbxax
Quadratic Quadratic Formula:Formula:
€
x =−b ± b2 − 4ac
2a
Notice where the letters come from for the formula
We use the quadratic formula when something can not be factored. However, it also works We use the quadratic formula when something can not be factored. However, it also works for factorable quadratic equations as well.for factorable quadratic equations as well.
You must get equation You must get equation equal to zero before you equal to zero before you
determine a, b, and c.determine a, b, and c.
2 6 5x x Ex. 1Ex. 1 Solve.Solve.
2 5 6 0x x 5
6
1
c
b
a
2( 5) ( 5) 4(1)
(1)
( )
2
6x
5 25 24
2x
5 1
2x
5 1
2x
3,2x
2 2 8 0x x Ex. 2Ex. 2 Solve.Solve.
2
8
1
c
b
a
2( 2) ( 2 (1)
(1)
() 4 )
2
8x
4, 2x
2 14 49 0x x Ex. 3Ex. 3 Solve.Solve.
14
4
1
9
b
a
c
2 (1)
(1
(14) (14) ( )
)
4
2
49x
7x
218 10 0x x Ex. 4Ex. 4 Solve.Solve.
10
18
1b
a
c
2 (14 8)
(
(1) (1)
82 1 )
(10)x
no solution
Ex. 5Ex. 5 Suppose a football player kicks a ball and gives it an Suppose a football player kicks a ball and gives it an initial upward velocity of 47ft/s. The starting height of initial upward velocity of 47ft/s. The starting height of the football is 3ft. If no one catches the football, how the football is 3ft. If no one catches the football, how
long will it be in the air?long will it be in the air?
NOTENOTE: For your homework tomorrow night, DO NOT answer the : For your homework tomorrow night, DO NOT answer the question. I want you to draw a sketch of what you think this picture question. I want you to draw a sketch of what you think this picture
would look like on a graph. Tell me what the x-axis symbolizes, what the would look like on a graph. Tell me what the x-axis symbolizes, what the y-axis symbolizes, and what the zeros represent in the problem.y-axis symbolizes, and what the zeros represent in the problem.
What does the x-axis stand for?What does the x-axis stand for?
What does the y-axis stand for?What does the y-axis stand for?
What do the zeros represent?What do the zeros represent?
Time the ball is in the airTime the ball is in the air
Height of the ballHeight of the ball
The amount of time it takes the ball The amount of time it takes the ball to hit the ground.to hit the ground.
01382 xx
13
8
1
c
b
a
a
acbbx
2
42
12
131488 2 x
2
128 x
3412
32 2
328 x
34x
Before you can find a,b, and Before you can find a,b, and c, you must get equation = c, you must get equation =
to 0.to 0.
Type what is under the Type what is under the radical exactly as written radical exactly as written
on your calculator.on your calculator.
Simplify the radicalSimplify the radical
Divide by the denominator, if you canDivide by the denominator, if you can
Roots:Roots:
053 2 x
5
0
3
c
b
a
a
acbbx
2
42
32
53400 2 x
6
600 x
6
152x
15460
152
3
15x
Simplify the radicalSimplify the radical
Divide by the denominator, if you Divide by the denominator, if you can. Since I can’t, divide by GCFcan. Since I can’t, divide by GCF
RootsRoots::
Solve the quadratic equation by using the Solve the quadratic equation by using the quadratic formula and leave answers in quadratic formula and leave answers in
simplest radical form.simplest radical form.
Solve the quadratic equation by Solve the quadratic equation by using the quadratic formula and using the quadratic formula and
round answers to the nearest tenth.round answers to the nearest tenth.
029122 xx
29
12
1
c
b
a
a
acbbx
2
42
12
29141212 2 x
2
2812 x
Since we are solving to the nearest Since we are solving to the nearest tenth, we do not need to simplify tenth, we do not need to simplify
the radical!the radical!
2
2812 x
2
2812 x
64.8x6.8x
35.3x4.3x
Be careful when typing this into Be careful when typing this into your calculator. I recommend that your calculator. I recommend that you type in the numerator then hit you type in the numerator then hit enter, then divide by denominator!enter, then divide by denominator!
Roots:Roots:
xx 1082
8
10
1
c
b
a
a
acbbx
2
42
12
8141010 2 x
2
6810 x
x10 x10
08102 xxSolve the quadratic equation by Solve the quadratic equation by using the quadratic formula and using the quadratic formula and
round answers to the nearest tenth.round answers to the nearest tenth.
2
6810x
2
6810 x
12.9x1.9x
87.x9.x
Roots:Roots:
Solve the quadratic equation by Solve the quadratic equation by using the quadratic formula and using the quadratic formula and
round answers to the nearest tenth.round answers to the nearest tenth.€
3x x − 2( ) =1
163 2 xx1 1
0163 2 xx
1
6
3
c
b
a
a
acbbx
2
42
32
13466 2 x
6
486x
6
486 x
6
486 x
15.2x1.2x
15.x2.x
RootsRoots
Solve the quadratic equation by using Solve the quadratic equation by using the quadratic formula and leave the quadratic formula and leave answers in simplest radical form.answers in simplest radical form.
xx 412 2 x4 x4
0142 2 xx
1
4
2
c
b
a
a
acbbx
2
42
22
12444 2 x
4
84x
248
224
224x
Divide by the denominator, if you Divide by the denominator, if you can. Since I can’t, divide by GCFcan. Since I can’t, divide by GCF
2
22 x
Roots:Roots:
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