[the discriminant][section 69]

(1.)  x2 + 2x + 8 = 0          here is the problem

b2 - 4ac              use the discriminant formula

=    (2)2 - 4(1)(8)       make substitutions

=     4 - 32               multiply

=    -28               subtract

result:  there are no real solutions

(2.)   3x2 - 3x - 4 = 0            here is the problem

b2 - 4ac             use the discriminant formula

=   (-3)2 - 4(3)(-4)        make substitutions

=    9 + 48               multiply

result:  There are two real solutions

(3.)   x2 = 5x + 5                here is the problem

-5x -5x            subtract 5x from each side
______________
x2 - 5x =   5           subtract

-5   -5        subtract 5 from each side
_________________
x2 - 5x - 5 = 0        subtract

b2 - 4ac           use the discriminant formula

=  (-5)2 - 4(1)(-5)     make substitutions

=   25 + 20              multiply

result:  there are two real solutions

(4.)  3x2 + 4x - (3/4) = 0         here is the problem

12x2 + 16x - 3 = 0      multiply thru by 4, cancel

b2 - 4ac     use the discriminant formula

=  (16)2 - 4(12)(-3)    make substitutions

=   256 + 144          multiply

result: There are two real solutions.

(5.)  20x = x2 + 100             here is the problem

x2 + 100 = 20x             just rearrange

-20x     -20x         subtract 20x from each side
__________________
x2 - 20x + 100 = 0       subtract

b2 - 4ac           use the discriminant formula

=  (-20)2 - 4(1)(100)        make substitutions

=    400 - 400              multiply

=    0                  subtract

result:  There is one real solution.

(6.)   4x2 + 2x - 9 = 0                here is the problem

b2 - 4ac             use the discriminant formula

=   (2)2 - 4(4)(-9)       make substitutions

=   4 + 144                 multiply

result: there are two real solutions

(7.)   -x2 + 7x - 15 = 0             here is the problem

b2 - 4ac          use the discriminant formula

=   (7)2 - 4(-1)(-15)    make substitutions

=      49 - 60                 multiply

=     -11                        subtract

result:  there are no real solutions

(8.)    (1/2)x2 - 16x + 132 = 0     here is the problem

x2 - 32x + 264 = 0     multiply thru by 2, cancel

b2 - 4ac            use the discriminant formula

=    (-32)2 - 4(1)(-264)             make substitutions

=     1024 + 1056                     multiply

result:  there are two real solutions

(9.)    x2 - 16 = 0                here is the problem

b2 - 4ac      use the discriminant formula

=       02 - 4(1)(-16)          make substitutions

=      0 + 64                     multiply

result:  there are two real solutions

(10.)   x2 - 4x + 4 = 0              here is the problem

b2 - 4ac          use the discriminant formula

=    (-4)2 - 4(1)(4)         make substitutions

=    16 - 16                   multiply

=     0                        subtract

result:  there is one real solution