[completing the square][section 66]
(1.) x2 + 4x = 12 here is the problem
[(1/2)(4)]2 = 4 square half the 2nd term
x2 + 4x + 4 = 12 +
4 add 4 to each side
(x + 2)2 = 16 factor, add
x + 2 = 4 x + 2 = -4 take square roots
-2 -2
- 2 -2 subtract 2 from each side
_______________ _____________
x = 2 ,
x = -6 subtract
results: x = 2 , x =
-6
(2.) x2 + 6x = 27 here is the problem
[(1/2)(6)]2 = 9 square half the 2nd term
x2 + 6x + 9 = 27 +
9 add 9 to each side
(x + 3)2 = 36 factor, add
x + 3 = 6 x + 3 = -6 take square roots
-3 -3
-3 -3 subtract 3 from each side
____________ _____________
x
= 3 , x = -9
subtract
results: x = 3 , x =
-9
(3.) n2 - 8n = -15 here is the problem
[(1/2)(-8)]2 = 16 square half the 2nd term
n2 - 8n + 16 = -15 +
16 add 16 to each side
(n - 4)2 = 1 factor, add
n - 4 = 1 n - 4 = -1 take square roots
+
4 +4 +
4 +4 add 4 to each side
___________ ___________
n =
5 , n =
3 add
results: n = 5 , n =
3
(4.) n2 - 2n - 8 = 0
+8 +8 add 8 to each side
__________________
n2 - 2n =
8 add
[(1/2)(-2)]2 = 1 square half the 2nd term
n2 - 2n + 1 = 8 + 1 add 1 to each side
(n - 1)2 = 9 factor, add
n - 1 = 3 n - 1 = -3 take square roots
+1 + 1 + 1
+1 add 1 to each side
_____________ ________________
n = 4 , n = -2
add
results:
n = 4 , n = -2
(5.) b2 + 10b = -16
[(1/2)(10)]2 = 25
square half the 2nd term
b2 + 10b + 25 = -16 +
25 add 25 to each side
(b + 5)2 = 9 factor, add
b + 5 = 3 b + 5 = -3 take square roots
-5
-5 -5 -5
subtract 5 from each side
___________ ______________
b = -2 , b =
-8 subtract
results: b = -2 , b =
-8
(6.) x2 - 16x = -60 here is the problem
[(1/2)(-16)]2 = 64 square half the 2nd term
x2 - 16x + 64 = -60 +
64 add 64 to each side
(x - 8)2 = 4 factor, add
x - 8 = 2 x - 8 = -2 take square roots
+ 8 +8 +
8 +8 add 8 to each side
____________________ ________________
x = 10 , x
= 6 add
results: x = 10 , x = 6
(7.) x2 - 10x = -24
[(1/2)(-10)]2
= 25 square half the 2nd term
x2 - 10x + 25 = -24 +
25 add 25 to each side
(x - 5)2 = 1 factor, add
x - 5 = 1 x - 5 = -1
take square roots
+ 5
+5 +5 +5 add
5 to each side
_________________ ________________
x = 6
, x = 4 add
results: x = 6 , x = 4
(8.) n2 + 6 = -5n here is the problem
+
5n + 5n
add 5n to each side
___________________
n2 + 5n + 6 = 0 add
[(1/2)(5)]2 = 6.25 square half the 2nd term
n2 + 5n + 6 + .25 =
.25 add .25 to each side
n2 + 5n + 6.25 =
.25 combine like terms
(n + 2.5)2 = .25 factor
n + 2.5 = 0.05 n + 2.5 = -0.05 take square roots
-2.5 -2.5
-2.5 -2.5 subtract 2.5 from
____________________ ___________________ each side
n = -2
n = -3
subtract
results: n = -2 , n = -3
(9.) a2 + 4a = 5 here is the problem
[(1/2)(4)]2 = 4
square half the 2nd term
a2 + 4a + 4 = 5 +
4 add 4 to each side
(a + 2)2 =
9 factor, add
a + 2 = 3 a + 2 = -3 take square roots
-2 -2
-2 -2 subtract 2 from each side
___________________
________________
a = 1
, a = -5 subtract
results: a = 1 , a = -5
(10.) x2 = 3x here is the problem
-3x -3x subtract 3x from each side
__________________
x2 - 3x = 0 subtract
[(1/2)(-3)]2 = 2.25 square half the 2nd term
x2 - 3x + 2.25 =
2.25 add 2.25 to each side
(x - 1.5)2 =
2.25 factor
x - 1.5 = 1.5 x - 1.5 = -1.5 take square roots
1.5 +1.5
+1.5 +1.5 add 1.5 to ea side
__________________ ________________
x =
3 , x = 0
add
results: x = 3 , x =
0