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x^2 ( 4x - 3)  (5x - 1)  =  0

Set each factor to 0 and solve

x^2   = 0     ⇒  x  = 0

4x - 3  =  0     add 3 to both sides

4x  = 3           divide both sides by 4

x  =  3/4

5x - 1  = 0      add 1 to both sides

5x  =  1     divide both sides by  5

x  =  1/5

4x^3  =  405x        divide both sides by 5

x^3  =  81x           subtract 81x  from both sides

x^3  -  81x  =  0        factor

x ( x^2  - 81)  =  0

x  (x - 9) (x + 9)  =  0     and we can write  this as

(x - 0 )  ( x - 9)  ( x + 9 )  =  0

The solutions  are

0,  9, -9

x^3 - 8x^2  - x  +  8  =  0       factor by grouping

x^2  (x  -  8)  -  1(x - 8)  =  0             ( x - 8 )  is the common factor

(x - 8)  (x^2  - 1)  =  0

(x - 8)  ( x + 1) ( x - 1)  =    0

Change the signs on the integers inside the parentheses  to get the solutions  of :

8 ,  -1,  1

x^2  +   18x  +  45  =  0

(x + 15) ( x + 3)   =  0

Once we have this form, change the signs on the integers inside the parentheses to get the roots 

⇒  -15, -3

We are  using  the property that

ax^m  *  bxⁿ    =   (a*b)  (x^m * xⁿ)  =  (a*b) (x^m+n )

So...for instance  .....  3x³ (7x²)  =  (3*7) (x³⁺²)  =  21x⁵

So......the last three are correct

f(x)  =  4sin(x)  +  6

g(x)   =  3x  +  5

Maximize    ( f °  g ) (x)

( f ° g) (x )  =     4 sin  ( 3x + 5)  +  6

Note that the maximum value  of sine  =  1

So......the max value  of   (f ° g)  (x)  =  

4 (1)  +   6   =

10

Note that  g(x)  doesn't really matter.....it's a "red herring"

(6x -  4)  (3x  - 1)    =

     6x  (3x  -  1)       =     18x^2  -  6x

+  -4    (3x  - 1)       =                 -12x  +  4

____________             ______________

                                     18x^2  -  18x  +  4

(x+2)(−3x^2+3x+1)     {  we can line this up just like regular addition as follows  }

    x (−3x^2+3x+1)     =       -3x^3  + 3x^2  +  x

+  2 (−3x^2+3x+1)      =                 -6x^2    + 6x   +  2

_______________             ____________________

                                            3x^3 - 3x^2  +  7x   +  2

Thanks!

Area  = 

   L     *      W   =

(x + 1)  (3x - 5)  =   

3x^2  + 3x  -  5x  -  5   =

3x^2  - 2x  - 5

If  f(x )  =  sec x  , tan (x)  = 1    and x  is in Quadrant I what is the value of  (f * f)(x)  ??

If  tan (x)  = 1,   then x  =  pi/4

And  

(f * f) (x)   =  f(x) * f(x)

So

(f * f) (pi/ 4)    =  f(pi/4) * f(pi/4)   =  sec(pi/4) * sec(pi/4)  =  √2 * √ 2   =   2

Note that  we are trying to minimze this sum  for   some value of r

S =   9 / (1 - r)  

If   r  =  -1/2       the sum  is 6

If r  =   -4/5        the sum is  5

If r  = -5/4          the sum is  4

But    l r  l    must be  <  1  ......so  r  =  -5/4  isn't allowed

So.....it appears that the sum is minimized  to an integer whenever  r  =  -4/5   and the minimized sum is  5

Tangent of the angle ABC =AC / AB.

Tan(58) =AC / 100

1.60 = AC / 100    cross multiply

AC = 1.60 x 100 = 160 feet - the width of the river.

Any radicals multiplied by each other can be simplified. Just combine them into one radical.

\(\sqrt{2}*\sqrt{7}*\sqrt{13}=\sqrt{2*7*13}\)

From there it is just simple multiplication.

\(\sqrt{2*7*13}=\sqrt{14*13}=\sqrt{182}\)

So, your answer is √182.

Hello and thanks for your response. I have one question though. 

f(x)=4sinx+6 not sin4x+6 wouldn't that make a difference? 

Thanks CPhill Again!

Scruffy has taken (3 - 4i)(1 + 5i) - (2 - i)

                        to  23 + 11i - (2 - i)

Finishing:              23 + 11i - (2 - i)  =  23 + 11i - 2 + i

                                                       =  23 - 2 + 11i + i

                                                       =  21 + 12i

Simplify.

3√(6√+15−−√)

-6i · sqrt( -44 )

Since sqrt( -44 )  =  i · sqrt( 44 )

                           =  i · sqrt( 4 ) · sqrt( 11 ) 

                           =  i · 2 · sqrt( 11 ) 

                           =  2i · sqrt( 11 ) 

-6i · sqrt( -44 )  =  -6i · 2i · sqrt( 11 ) 

                         =  -12 · i² · sqrt( 11 ) 

Since  i²  =  -1:

                         =  -12 · -1 · sqrt( 11 )

                         =  12· sqrt( 11 )

Simplify.

(7−2i)(8+9i)

Menards has a 90 yard roll of chain for sale.  A tow truck driver needs to purchase 60 feet to pull cars that are disabled.  A dog breeder needs 5 chains that are 16 feet each.  Then an artist comes in and buys some chain.  After his purchase, 1/3 of the original roll is left.  How much did the artist purchase, in yards, to the nearest tenth?

90 yards x 3 feet = 270 feet.

270 x 1/3 = 90 feet left

270 - 90 = 180 feet sold

60 feet are sold to tow truck driver.

5 x 16 = 80 feet sold to the dog breeder

60 + 80 = 140 feet sold to dog breeder and tow truck driver.

180 - 140 =40 feet sold to the artist

40/3 =13.3 yards sold to the artist.

Simplify:  sqrt( 4x² / 3y )

                sqrt( 4x² / 3y )  =  sqrt( 4x² ) / sqrt( 3y )

But  sqrt( 4x² )  =  2·| x |                                  Technically, you need to use absolute value bars.

               sqrt( 4x² / 3y )  =  2·| x |  / sqrt( 3y )

Now, multiply both the numerator and the denominator by sqrt( 3y )

              sqrt( 4x² / 3y )  =  [ 2·| x | · sqrt( 3y) ] / [ sqrt( 3y ) · sqrt( 3y) ]

Since  [ sqrt( 3y ) · sqrt( 3y) ]  =  3y

               sqrt( 4x² / 3y )   =  [ 2·| x | · sqrt( 3y) ] / ( 3y )

Now it is correct.

Jon has 3 boards that are each 8 feet long.  First he makes 3 identical shelves.  Then he uses 18 inches to make a bird house.  When finished, he has 25% of the original amount left.  How much wood was used for each shelf, in feet?

3 x 8 =24 feet - That Jon started with.

25% x 24 = 6 feet left.

24 - 6 = 18 feet used to make 3 shelves and a bird house

18 inches / 12 = 1.5 feet used for the bird house.

18 - 1.5 = 16.5 feet used for the 3 shelves

16.5 / 3 = 5.5 feet used for each of the 3 shelves.

Multiply.

3√⋅5√⋅7√

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