CPhill

This is pretty simple....believe it, or not    !!!

1.   x^2  +  12x

Take  1/2  of 12  =  6

Square this  = 6^2  =  36.....and that's what we need to add to complete the square  !!!!

2. x^2 - 8x

Take (1/2)  of -8  = -4

Square this  = (-4)^2  =   16

3. x^2  - 18x

Take (1/2) of -18  = -9

Square this  (-9)^2   =  81

log (2x)  + log (x -5)    = 2

We can use this rule to simplify things, Manuel

log a  + log b  =  log (a * b)....so ....we can write the left sides as

log[ (2x * (x -5) ]  = 2

log [  2x^2 - 10x]  =  2      in exponential form we have

10^2  = 2x^2 - 10x

100  =  2x^2  - 10x        rearrange as

2x^2 - 10x - 100  = 0       divide through by 2

x^2 - 5x -50  =  0        factor as

(x - 10)(x + 5)  = 0

Setting each factor to 0 and solve for x  and we have that

x  = 10     or   x  = -5

We must reject  x  = -5   because it would mean that we would be taking logs of negative numbers in the original equation

So.....x  = 10

Let x  be the smaller dimension of the window.....then 1.2 x  is the other dimension

So we have

 x * (1.2x)  = 811.2

1.2x^2  = 811.2       divide both sides by 1.2

x^2  = 676       take the square root of both sides

x  = 26 in    this is the smaller dimension

1.2 (26)  =  31.2 in      this is the larger dimension

So.....  "a"  is correct

None of the answers are fully factored

"B"   is correct...however......(3x - 6y)(3x - 6y)   can be factored further as

3(x - 2y) * 3 ( x - 2y)  =  9 (x -2y)² 

9x^2 -36xy + 36y^2

Take out the greatest common factor

9 (x^2 - 4xy + 4y^2)

Factor the polynomial in theparenthese

9 (x - 2y) (x - 2y)

9 (x - 2y)²

The grey  half circle on the left  has a radius  of 12cm

Its area is  (1/2)pi(12)^2   =   72pi  cm^2 ≈  226.2 cm^2    (1)

The remaining area is a rectangle  with dimensions 12 * 24  =   288 cm^2  (2)

So  the approximate total area  is  288 + 226.2  =   514.2  cm^2  

The approximate area  of the two black pieces  is  (2) - (1)   =  61.8 cm^2

4^x-2  = 2^x

The idea here is to make the bases equal  and then solve for the exponents

(2²)^x-2  = 2^x

(2)^[2(x-2)]  = 2^x

(2)^{2x - 4}  = 2^x

The bases are the same....solve for the exponents

2x - 4  = x        add 4 to both sides.....subtract x from both sides

x  =  4

Look at  "D"  ⇒ Expression 4

³ √ [ 343 x⁹ y z⁴  ]  =

³√343   * ³√x9   *  ³√y   * ³√z⁴   =

7  * x³  * y^1/3  * z^4/3 

a.  We have three half-circles each with a radius  of   6

So....the total crcumference is

3  (1/2) * 2 * pi  (6)   =    18 pi  cm  ≈  56.55 cm

b.  This is an equilateral triangle with sides of 12

Its area is

(1/2)12^2* √3/2   =   144√3/4   =  36√3 cm^2   ≈  62.35 cm ^2    (1)

c. The total area of the three half-circles is

3 (1/2) * pi (6)^2   =  54 pi  cm^2 ≈  169.65 cm^2   (2)

So....the approximate area is  (1) + (2)  ≈  232 cm^2

10000  =    10000 /  [ 1 + 9e^(-1.13t) ]

Divide both sides by 10000

1  =    1 /  [ 1 + 9e^(-1.13t) ]

 1 + 9e^(-1.13t)   =  1

9 e^(-1.13t)  =  0

e^(-1.13t)  =  0

Note that an exponential  can never  = 0

So.....it will  never actually  =  10000

The sum of 5 and √5  is irrational

The product of 5 and 5√5  is irrational

The product of 5 and √5  is irrational

Any irrational number multiplied by a non-zero rational number is equal to an irrational number