CPhill

(1.496 x 10^11) / (5.2 x 10^-3)   we can write this as :

[1.496 / 5.2 ]   * [ 10^11 * 10^3  ]  ≈

.2877   *    10^14  ≈

2.877 * 10^13

Just input it normally.......the calculator will handle all the basic operations with fractions

2x^2 + 5x + b  = 0

If b is a positive integer.......it cannot be an integer larger than 3 because the discriminant  would be < 0

So......the only possible positive integer values for b are either 1, 2 or 3

If b =  1, the discriminant   =  √17 and this will  not  produce a rational  solution

If b = 2, the discriminant = √9  = 3    and this will produce a rational  solution

And if b = 3, the discriminant   = √1   = 1   and this too will produce a rational solution

So....2,3 produce rational roots for b....where b is a positive integer

And the sum of these  = 5

Sorry....it should be   3/4 + √3  =   [ 3 + 4√3 ] / 4

sin pi/3 =   √3/2    so the square of this is 3/4

tan pi/3   = √3/1     

So  √3/4  + √3/1   =  [ √3 + 4√3] / 4   =  [5√3] / 4

9x^2 -18x - 16  = 0

15x^2 + 28x + 12 = 0     

Since both quadratics = 0, set them = to each other

9x^2 -18x - 16  = 15x^2 + 28x + 12       subtract 9x^2 from both sides, add 18x, 16 to both sides

6x^2 + 46x + 28  = 0     divide everything by 2

3x^2 + 23x + 14   = 0     factor

(3x + 2) (x + 7)   = 0     setting both factors to 0 and solving,  x = -2/3   or x = -7

However since we're trying to find where both functions = 0........the solution is x = -2/3

The second solution of x = -7 is another intersection point of both graphs

f(x) =  (1/4) x + 3      write f(x)  as y

y = (1/4)x + 3      the object is to get x by itself and then  "exchange" x and y

Subtract 3 from both sides

y - 3  = (1/4)x     multiply both sides by 4

4y - 12 = x        "exchange"   x and y

4x - 12   = y

For y, write f^-1 (x)

4x - 12  =  f^-1(x)   = 4(x - 3)    =   "C"

6^(x+8)=12      take the log of both sides

log 6^(x + 8)  = log 12       and by a  log property we can write

(x + 8) log 6   = log 12       divide both sides by log 6

(x + 8)    = [ log 12 / log 6]       subtract 8 from each side

x  = [ log12 / log 6 ]  -   8    =  about    -6.613

We can solve this :

20000(1 + r)^2   + 20000(1 + r)   + 20000   = 66200     subtract  66200 from each side

20000 (1 + r)^2 + 20000 (1 + r)  - 46200     =  0         let  1 + r  = x

20000x^2 + 20000x  - 46200  = 0       divide through by 20000

x^2 + x - 231/100  =0         add 231/10  to both sides and completet the square on x

x^2 + x + 1/4 =   231/100 + 1/4      factor and simplify

(x + 1/2) ^2   =  256/100      take the positive square root of both sides

x + 1/2 =  sqrt(256/10)

x + 1/2  = 16/10

x + 1/ 2 =    8/5

x = [ 8/5 - 1/2 ]  =     11/10   = 1.1

So.....  1 + r  = 1.10      subtract 1 from both sides

r = .10   =  10%

f a ship's path is mapped on a coordinate grid, it follows a straight-line path of slope 3 and passes through point (2, 5).

Part A: What is the equation of the path?

Part B: Does the ship pass through point (7, 22)?

Part C: A second ship follows a straight line, with the equation x + 3y − 6 = 0. Are these two ships sailing perpendicular to each other?

The equation of the first line is given by :

y - 5 = 3(x - 2)     simplify

y = 3x - 6 + 5

y = 3x - 1

For part B, substitute  7 for x in the equation and we have :  3(7) - 1  =  21 - 1  = 20......so......the ship passes through (7, 20), but not (7,22)

Part C......x + 3y - 6  = 0   .....rearrange this as      y = (-1/3)x + 2..........the slope of this line is the negative reciprocal of the first line, so they are perpendicular.......