geno3141

x² - 2x + 5  =  0

Using the quadratic formula: 

x₁  =  [ 2 + sqrt( (-2)² - 4(1)(5) ) ]  /  [ (2)(1) ]  =  [ 2 + sqrt( 4 - 20) ]  / 2  =  [ 2 + sqrt(-16) ]  / 2  =  ( 2 + 4i ) / 2  =  1 + 2i

x₂  =  [ 2 - sqrt( (-2)² - 4(1)(5) ) ]  /  [ (2)(1) ]  =  [ 2 - sqrt( 4 - 20) ]  / 2  =  [ 2 - sqrt(-16) ]  / 2  =  ( 2 - 4i ) / 2  =  1 - 2i

a  =  1 / ( 1 + 2i )

b  =  1 / ( 1 - 2i )

a·b  =  [ 1 / ( 1 + 2i ) ]  ·  [ 1 / ( 1 - 2i ) ]  =  1 / ( 1² - (2i)² )  =  1 / ( 1 + 4 )  =  1 / 5

2x² - 5x - 4  =  2x + 3     --->     2x² - 7x - 7  =  0

Using the quadratic formula with  a = 7  b = -7  and  c = -7:

x  =  [ - -7  +/-  sqrt( (-7)² - 4(2)(-7) ) ]  /  [ (2)(2)

x  =  [ 7  +/-  sqrt(105) ]  /  4 

    0.7266666666...

+  0.0233333333...

=  0.7499999999...

=  0.75

=  3/4

sqrt(28x) · sqrt(15x) · sqrt(27x)  =  sqrt(4·7x) · sqrt(3·5x) · sqrt(9·3x)

                                                   =  2 · sqrt(7x) · sqrt(3·5x) · 3 · sqrt(3x)

                                                   =  6 · sqrt(7x · 3 · 5x · 3x)

                                                   =  6 · 3 · x · sqrt(7 · 5 · x)

                                                   =  18·x·sqrt(35·x)

10·pi + 10·sqrt(2)  is approximately equal to  45.558

The problem becomes:  how many integers are in the range  -45, -44, ..., -1, 0, 1, ..., 44, 45?

If the order in which they are chosen is not important, there are 3 marbles chosen from a possible 8:

             ₈C₃  =  56

The prime numbers are 2, 3, and 5.

The numbers that aren't prime are 1, 4, and 6.

The probability that the first die shows a prime number is:  3/6.

The probability that the second die shows a prime number is:  3/6.

The probability that both show prime numbers is:  3/6 x 3/6  =  ...

10!  =  10 x 9 x 8 x 7 x 6 x 5 x 4 x 3 x 2 x 1

27  =  3 x 3 x 3 

When you divide 10! by 27 (and cancel out the corresponding 3s) you get:  10 x 8 x 7 x 2 x 5 x 4 x 3 x 1 

with no remainder; remainder = 0.

Take the number of people who own cats.

Add to this, the number of people who own dogs.

From this total, subtract the number of people who own both cats and dogs.

n = 0, 1, 4

I beleive that these are the only correct answers.