geno3141

Factor:  5x² - 25x

Begin by factoring out the common term 5x

     5x² - 25x  =  5x(x - 5)

This is all that can be done in this problem.

Infinity isn't a number, so it doesn't follow the rules of numbers.

Also, infinity comes in various sizes. Therefore, one infinitely large set may be the same size as another infinitely large set; but, then again, one infinitely large set may be larger (or smaller) than another infinitely large set.

5^{(x + 1) / (1 - 2x)}  <  0.3^-3

5^{(x + 1) / (1 - 2x)}  <  125

5^{(x + 1) / (1 - 2x)}  <  5³

(x + 1) / (1 - 2x)  < 3

(x + 1)   <   3(1 - 2x)

x + 1   <   3 - 6x

7x   <   2

x   <   2/7

Solve:  2^3x-2 - 2^3x-3 - 2^3x-4 - 4  =  0

2^3x-2  =  2^3x · 2^-2  =  2^3x · (1/4)  =  2^3x / 4

Similarly,

2^3x-3  =  2^3x / 8

2^3x-4  =  2^3x / 16

So:    2^3x-2 - 2^3x-3 - 2^3x-4 - 4  =  0     --->     2^3x / 4  -  2^3x / 8  -  2^3x / 16  -  4  =  0

Mulitply both sides by 16:                --->   16 [ 2^3x / 4  -  2^3x / 8  -  2^3x / 16  - 4 ]  =  16 [ 0 ]

                                                        --->        4 · 2^3x  -  2 · 2^3x  -  1 · 2^3x  -  64  =  0

Add 64 to both sides:                      --->        4 · 2^3x  -  8 · 2^3x  -  1 · 2^3x   =   64

Factor out 2^3x:                                --->        2^3x [ 4 - 2 - 1 ]   =   64

Simplify:                                          --->        2^3x [ 1 ]   =   64

Simplify:                                          --->        2^3x   =   64

Rewrite 64:                                     --->        2^3x   =   2⁶

                                                       --->          3x   =   6

                                                       --->              x  =  2

(But, this isn't a quadratic equation!)

If each way station needs 10 plants and 65% of the seeds planted sprout, to find the number of seeds need to produce 10 plants:

     0.65 · Number  =  10

--->         Number  =  10 / 0.65     =     15.4     =     16 seeds

For 15 way stations:  15 x 16 seeds  =  240 seeds

How can you find the area of a triangular prism?

If the prism is an equilateral prism (each of its four triangular sides is an equilateral triangle), then you can use this formula:

Total Surface Area  =  sqrt(3) · (length of a side)²

If it isn't an equilateral prism, find the area of each side and add them together.

How can you find the radius of a circle when you know its radius?

A  =  pi · r² 

Divide the Area by pi:

A / pi  =  r²

Find the square root of that answer:

r  =  sqrt( A / pi )

The amount of time that it takes a population to triple if the growth rate is 3.9%:

Use the compound interest formula:  A  =  Pe^r·t

A  =  final amount         P  =  initial amount          r  =  rate (as a decimal)         t  =  time

--->   Let the initial amount = x, then the final amount will be 3x         r  =  0.039         time = t

--->     3x  =  x·e^0.039·t  

Divide both sides by x:

--->     3  =  e^0.039·t  

Take the ln of both sides:

--->     ln(3)  =  ln( e^0.039·t )

Using the appropriate property of logs:

--->     ln(3)  =  0.039 · t · ln( e )

Since the ln(e)  =  1:

--->     ln(3)  =  0.039·t

Divide both sides by 0.039:

--->     t  =  ln(3) / 0.039  =  28.17 years (approximately)  or  28 years, 2 months

8(x - 4)  =  x - 12

Multiply the left side, using the distributive property:

8x - 32  =  x - 12

Subtract x from both sides:

7x - 32  =  -12

Add 32 to both sides:

7x  =  20

Divide both sides by 7:

x  =  20/7

You start with 500 sheets of notebook paper.

After 1 week, you have 72% of the sheets left.

--  Since 72% of 500 =  0.72 x 500 =  360, you have 360 sheets left.

You use the same number of sheets each week.

--  Since you started with 500 sheets and you have 360 sheets left after one week, you have used 500 - 360 = 140 sheets the first week.

You use the same number of sheets each week, so you use 140 sheets each week.

Write an equation that represents the number of sheets remaining (y) after x weeks.

y  =  500 - 140x

In slope-intercept form, this becomes:  y  =  -140x + 500.