[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