Answers

©
9 -6
______________________________________________ D ATE
_ _ _ _ _ _ _ _ _ _ _ _ P E R IO D
N A ME
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9 -6
Study Guide and Intervention
F a c t o r P erf ec t S q u a re T rino m ia ls
P e rfe c t S q u are T rin o mial
a trinomial of th e form
a2
1 2ab 1
b2
or
a2
2 2ab 1
b2
The patterns shown below can be used to factor perfect square trinomials.
S q u arin g a B in o mial
(a 1
4)2
a2
5
1 2(a)(4) 1
5 a 2 1 8a 1 1 6
42
(2x 2 3)2 5 (2x )2 22(2x )(3) 1 32
5 4x 2 2 1 2x 1 9
F ac to rin g a P e rfe c t S q u are T rin o mial
a2
a2
1 8a 1 1 6 5
1 2(a)(4) 1
5 (a 1 4)2
S q u are R o o t P ro p e rty
Example
Example 2 Factor 16x2 2 32x 1 15.
S ince 15 is not a perfect square, use a different
factoring pattern.
16x2 2 32x 1 15
5 16x2 1 mx 1 nx 1 15
5 16x2 2 12x 2 20x 1 15
5 (16x2 2 12x) 2 (20x 2 15)
5 4x(4x 2 3) 2 5(4x 2 3)
5 (4x 2 5)(4x 2 3)
O rig inal trinomial
W rite th e p attern.
m 5 21 2 and n 5 220
Grou p terms .
F ind th e GC F .
F actor by g rou p ing .
2(7x 1 10 y )(7x 2 10 y )
7 . 25c2 2 10c 2 1
p rime
Glencoe Algebra 1
10. 16m2 1 48m 1 36
4(2m 1 3)2
13. 36x2 2 12x 1 1
(6x 2 1)2
©
Glencoe/McGraw-Hill
S olv e each equation. C heck y our solutions.
b . ( a 2 5) 2 5 64
O rig inal eq u ation
(a 2 5)2 5 64
a 2 5 5 6Ï64
w
S q u are R oot P rop erty
a 2 5 5 68
6458?8
a5568
A d d 5 to each s id e.
a 5 5 1 8 or a 5 5 2 8 S ep arate into 2 eq u ations .
a 5 13
a 5 23
S olv e each eq u ation.
The solution set is {23, 13}. S ince (23 2 5)2 5 64 and (13 2 5)2 5 64, the solutions check .
Exercises
S olv e each equation. C heck y our solutions.
2. m2 1 10m 1 25
ye s; (m 1 5)(m 1 5)
3. p2 1 8p 1 64
no
5. x2 1 22x 1 121
(x 1
11)2
8 . 169 2 26r 1 r2
(13 2 r )2
11. 16 2 25a2
(4 1 5a)(4 2 5a)
14. 16a2 2 40ab 1 25b2
(4a 2 5b)2
553
6. 81 1 18s 1 s2
(9 1
s)2
5
1
2
6
5 15 6
1. x2 1 4x 1 4 5 0 {22}
2. 16n2 1 16n 1 4 5 0 2 }
4. x2 1 10x 1 25 5 0 {25}
5. 9x2 2 6x 1 1 5 0 }
6. x2 1 x 1 } 5 0 2 }
8 . p2 1 2p 1 1 5 49
9. x2 1 4x 1 4 5 64
5
Factor each poly nomial if possib le. If the poly nomial cannot b e factored , write
p r im e .
4. 98x2 2 200y2
F or any nu mber n . 0 , if x 2 5 n, th en x 5 6Ïn
w.
a. x2 2 6x 1 9 5 0
x2 2 6x 1 9 5 0 O rig inal eq u ation
x2 2 2(3x) 1 32 5 0 R ecog niz e a p erfect s q u are trinomial.
(x 2 3)(x 2 3) 5 0 F actor th e p erfect s q u are trinomial.
x 2 3 5 0 S et rep eated factor eq u al to 0 .
x 5 3 S olv e.
The solution set is {3}. S ince 32 2 6(3) 1 9 5 0, the solution check s.
Determine whether each trinomial is a perfect square trinomial. If so, factor it.
ye s; (x 2 8)(x 2 8)
(continued)
2
5
7 . 25k2 1 20k 1 4 5 0 2 }
5 13 6
6
3. 25d2 2 10d 1 1 5 0 }
5
1
4
{28, 6}
(7x 2 2)(x 2 1)
12. b2 2 16b 1 256
p rime
15. 8m3 2 64m
5
14. ( y 1 6)2 5 1 {27, 25}
16. (2x 1 1)2 5 1 {21, 0 }
17 . (4x 1 3)2 5 25 22, }
19. (x 1 1)2 5 7
20. ( y 2 3)2 5 6
{21 6 Ï7w }
Glencoe A lg ebra 1
©
5
{3 6 Ï6w }
554
Glencoe/McGraw-Hill
Answers
1
4
12. 16y2 1 8y 1 1 5 0 2 }
13. (x 1 3)2 5 49 {210 , 4}
8m(m 2 2 8)
6
{210 , 6}
10. x2 2 6x 1 9 5 25 {22, 8} 11. a2 1 8a 1 16 5 1
{23, 25}
9. 7x2 2 9x 1 2
1
2
1
2
6
6
15. (m 2 7)2 5 49 {0 , 14}
5 43 6
18 . (3h 2 2)2 5 4 } , 0
21. (m 2 2)2 5 5
{2 6 Ï5w }
Glencoe A lg ebra 1
(L es s on 9-6 )
Therefore 16x2 2 32x 1 15 5 (4x 2 5)(4x 2 3).
Exercises
1. x2 2 16x 1 64
Study Guide and Intervention
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An sw e rs
A17
16n2 2 24n 1 9 5 (4n)2 2 2(4n)(3) 1 32
5 (4n 2 3)2
_ _ _ _ _ _ _ _ _ _ _ _ P E R IO D
S o lv e E q u a t io ns wit h P erf ec t S q u a res F actoring and the Z ero P roduct P roperty
can be used to solv e equations that inv olv e repeated factors. The repeated factor g iv es just
one solution to the equation. Y ou may also be able to use the square root property below
to solv e certain equations.
42
4x 2 2 1 2x 1 9 5 (2x )2 22(2x )(3) 1 32
5 (2x 2 3)2
Example 1 Determine whether
16n2 2 24n 1 9 is a perfect square
trinomial. If so, factor it.
S ince 16n2 5 (4n)(4n), the first term is
a perfect square.
S ince 9 5 3 ? 3, the last term is a
perfect square.
The middle term is equal to 2(4n)(3).
Therefore, 16n2 2 24n 1 9 is a perfect
square trinomial.
______________________________________________ D ATE
P e rfe c t S q u are s an d F ac to rin g
P e rfe c t S q u are s an d F ac to rin g
L esso n 9 -6
Glencoe/McGraw-Hill
N A ME

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