geno3141

Since the center of the circle is (-1,1) and a point on the circle is (1,2), the radius of the circle can be found by finding the distance from the center to the point on the circle.

distance  =  sqrt( (x₂ - x₁)² + (y₂ - y₁)²​ )  =  sqrt( (1 - -1)² + (2 - 1)² )  =  sqrt( 2² + 1² )  =  sqrt( 4 + 1)  =  sqrt( 5 )

If the center of the circle is (h, k) and the radius is r, the tormula for the equation of the circle:  

     (x - h)² + (y - k²)  =  r²   --->     (x - -1)² + (y - 1)²  =  5

     --->     (x + 1)² + (y - 1)² =  5

     --->     [ x² + 2x + 1 ] + [ y² - 2y + 1 ]  =  5

     --->     x² + y² + 2x - 2y - 3  =  0

A = 1, B = 2, C = -2, D = -3

A + B + C + D  =  -2

Problem:  x² - 8  =  sqrt(4·x²)

     --->     x² - 8  =  sqrt(4) · sqrt(x²)

     --->     x² - 8  =  2· | x |

Split the absolute value into two parts:

Case 1:  If x >= 0, then | x | = x

     --->     x² - 8  =  2· | x |

     --->     x² - 8  =  2 · x

     --->     x² - 8  =  2x

     --->     x² - 2x - 8  =  0

     --->     (x - 4)(x + 2)  =  0

     --->     x = 4  or  x = -2

     --->     Since this case assumes that x >= 0, keep only the value  x = 4.

Case 2:  If x < 0, then | x | = -x

     --->     x² - 8  =  2· | x |

     --->     x² - 8  =  2 · -x

     --->     x² - 8  =  -2x

     --->     x² + 2x - 8  =  0

     --->     (x + 4)(x - 2)  =  0

     --->     x = -4  or  x = 2

     --->     Since this case assumes that x < 0, keep only the value  x = -4.

These are the two solutions.

A log is an exponent.

For example  log₂8 = 3  because 2³ = 8.

                      log₂8 = x  can be converted to  2^x = 8  (where x becomes the exponent)

    so, the answer to the log problem (the 'x') becomes the exponent.

If  f(x)  =  x² - 2x,  find all real numbers x such that  f(x)  =  f( f(x) ).

f( f(x) )  =  f( x² - 2x )  =  ( x² - 2x )²  - 2( x² - 2x )

Since  f(x)  =  f( f(x) )     --->     x² - 2x  =  ( x² - 2x )²  - 2( x² - 2x )

                                      --->     x² - 2x + 2( x² - 2x )  =  ( x² - 2x )² 

                                      --->   3( x² - 2x )  =  ( x² - 2x )² 

Case 1:  If x² - 2x  <>  0     --->   3( x² - 2x )  =  ( x² - 2x )² 

                  divide both sides by x² - 2x     --->     3  =  x² - 2x

                                                                  --->     x² - 2x - 3  =  0

                                                                  --->     (x - 3)(x + 1)  =  0

                                                                  --->     x = 3     or     x = -1

Case 2:  If x² - 2x  =  0     --->     x(x - 2)   0

                                         --->     x = 0     or     x = 2

These are the four possible answers.

Let cos(Ø) = x/r.

Let sin(Ø) = y/r

Since sin(Ø) = sqrt(2) / 6     --->     y = sqrt(2)     and     r = 6.

By the Pythagorean Identity,  x² + y²  =  r²     --->     x² + ( sqrt(2) )²  =  6²     --->     x² + 2  =  34     --->     x  =  sqrrt(34)

Therefore  cos(Ø)  =  sqrt(34) / 6

This is a "combinations" problem  --  choosing an unordered group of 2 from a group of 5.

C(5, 2)  =  5! / ( 2! x 3! )  =  10

The probabiltity that she did not pass the second exam.

In order to take the second exam, she must have passes the first.

So, in order to pass the first exam and to fail the second:  0.9 x 0.2  =  0.18.

After failing the second exam, she is not allowed to take the third exam.

I believe that the answer is 0.18.

You won't notice anything unusual about the first song.

However, if the second song comes from the same album (11/320 -- including possibly a repeat of the first song), you might wonder about that.

Getting the third song from the same album:  11/320 x 11/320.

Getting the fourth song from the same album:  11/320 x 11/320 x 11/320.

Getting the fifth song from the same album:  11/320 x 11/320 x 11/320 x 11/320  =  1.396 x 10^-6  =  0.00 00 014

This is about 14 out of 10,000,000  =  1 out of 700,000  (approximately).

Even though this is a slim chance, this is a lot better -- a lot, lot, lot, lot better -- than winning the lottery.

This is an exponential function:  y  =  2^x

It isn't linear, because:  although the x values increase by the same amount, the y-values don't.

It is exponential because as the x values increase by the same amount, the y-values are each 2 times the value of the preceding y-value.

cos(135^o)  =  - sqrt(2) / 2