CPhill

(2*sqrt(2)*e^t)^2

Notice that   2sqrt(2)   = sqrt (4 * 2)   =  sqrt (8)

So we have

[ sqrt (8) * e^t ] ^2   =

[sqrt (8) ]² * [ e^t ]²  =

8 * e^2t

Assuming that you know the x and y coordinates of two points on the line ....the midpoint is given by :

Add  the x coordinates and divide by 2     =    x midpoint coordinate

Add the y coordinates and divide by  2  =  y midpoint coordinate

Example 

(5, 6)   and  ( 7, 8)

Midpoint  =   [  ( 5 + 7) / 2  ,  (6 + 8) / 2 ]  =  [ 12/ 2,  14/2 ]    =  [ 6, 7 ]

x^2 - x  - 12  = 0

Here's the way I do it .....this method applies only if the lead coefficient is an understood "1"...as in this case........

[ there are also other methods ]

Write this as

x^2  - 1x   - 12  = 0  

Now......   look  at  the term on  "x"   = -1

And look at the constant term  = -12

Ok....we  are looking for two integers that  multiply to  - 12   and  that  sum to - 1  

Possibilities

1   -12

-1   12

2    -6

-2    6

3   - 4

-3    4

Notice that the only thing that works is   3  and - 4

(3) + ( - 4)    =  -1        and

(3) * (-4)  = -12

So....we just factor with these...i.e.,

(x  - 4)  ( x + 3)  = 0

Now.....set each factor to 0 and solve for  x....so we have.....

x  - 4  = 0                          and                        x + 3  = 0

add 4 to both sides                                    subtract 3 from both sides

x  = 4                                                             x  = -3

And those are the two solutions

The "smaller" x  = -3

The "larger" x  = 4

Does that help  ????

Try this one for yourself :      x^2  + 5x + 6  = 0 

The last choice is the correct one......

Correct, NSS !!!!!

I'm assuming that you have

 (5+2.5sqrt(3))/(25/4*sqrt(3) )   =

5 / (25/4*sqrt(3) )    +  2.5sqrt (3) / (25/4 * sqrt (3) )  =

5 / (6.25 sqrt (3) )   +  2.5sqrt (3) / ( 6.25 * sqrt (3) )   =

5 / (6.25 sqrt (3) )   +  2.5 / ( 6.25  ) 

.8 / sqrt (3)   +  .4      =        rationalize the denominator in the frist fraction

.8sqrt (3) / [ sqrt (3) * sqrt (3) ]  +   .4  =

.8 sqrt (3) / 3  +   .4   =              get a common denominator

[.8sqrt (3)  + .4 (3)]  / 3  =

 [ .8 sqrt (3)  + 1.2 ]  / 3  =

.4 [ 2sqrt (3)  + 3 ]  / 3  =

(.4/3) [ 2sqrt (3)  + 3 ]  =

(4/30) [ 2sqrt (3) + 3 ] =

(2/15) [ 2sqrt (3) + 3  ]  ≈  .861880

Writing the vector components thusly :

A  = < 2cos 0° , 2sin 0° >

B  =  < 3.75 cos 90°, 3.75 sin 90° >

C  =  < 2.5 cos 180°. 2.5 sin 180° >

D = < 3 cos 270°, 3 sin 270° >

Sum the x components  = 2cos 0° + 3.75cos 90° + 2.5cos 180° + 3cos 270°  = 2 + 0 - 2.5 + 0  =  -.5

Sum the y components  = 2sin 0° + 3.75sin 90° +  2.5 sin 180° + 3 sin 270° = 0 + 3.75 + 0 - 3  = .75

Resultant vector  =  < x , y >  =  < -.5, .75 >     →  in the second quadrant

Resultant magnitude  = sqrt ( x^2 + y^2 )  =  sqrt [ (-.5)^2  + (.75)^2 ] ≈  . 90134 km

Direction  =   tan^-1 (y/x)  = tan^-1 ( .75 / -.5)  = tan^-1 ( -3/2) = tan^-1 (-1.5)  ≈ -56.31°

However....this is a second quadrant angle so, with respect to due west, it is 56.31° north of due west 

[i.e.  123.69°  in standard position  ]

C.   Its has only one pair of congruent sides

14 + 2g  = 20         subtract 14 from both sides

2g  = 6               divide both sides by 2

g  = 3    =    [games that Blue Jays lost ]

Trapezoid