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\(\ \phantom{=}\,\dfrac{\frac{u+4}{25}-\frac{u+4}{5}}{\frac{4}{u+4}}\\~\\ =\,(\frac{u+4}{25}-\frac{u+4}{5})\div\frac{4}{u+4}\\~\\ =\,(\frac{u+4}{25}-\frac{u+4}{5})\cdot\frac{u+4}{4}\\~\\ =\,(\frac{u+4}{25}-\frac{u+4}{5}\cdot\frac55)\cdot\frac{u+4}{4}\\~\\ =\,(\frac{u+4}{25}-\frac{5(u+4)}{25})\cdot\frac{u+4}{4}\\~\\ =\,(\frac{1(u+4)-5(u+4)}{25})\cdot\frac{u+4}{4}\\~\\ =\,(\frac{-4(u+4)}{25})\cdot\frac{u+4}{4}\\~\\ =\,\frac{-4(u+4)(u+4)}{4(25)}\\~\\ =\,\frac{-(u+4)(u+4)}{25}\\~\\ =\,\frac{-(u+4)^2}{25} \)

Here is another way:

Area =Base x Height / 2

Area =8 x 6/2 =24 units^2

Perimeter =6 + 8+ 10 =24 units

Semi-perimeter =24/2 = 12 units.

Area = Semi -perimeter x inradius

Inradius =Area / Semi-perimeter

Inradius =24 / 12 =2 units.

We can use a little logic, here

Let the sides  of the triangle be  AB, BC, AC

We have a 6-8-10  right triangle

The  area  =  (1/2)  [ Product of the legs ]  =  (1/2) [ 8 * 6 ]  = 24

Let the center of the incircle  = O

We can divide the right triangle into  three smaller triangles  =  AOB, BOC and AOC

The inradius will be tangent to each side of triangle ABC, so it will form an altitude to each of the three smaller triangles

So.....the area of ABC  = (1/2) (inradius) [ AB + BC + AC]

24   =  (1/2) (inradius) [ AB + BC + AC ]

24 / (1/2) =  (inradius)  [ perimeter ]

48  =  (inradius)  [ 6 + 8 + 10 ]

48  = (inradius) [ 24 ]

inradius  =  48 / 24   =   2 units

This is a little easier than you might realize

Let us call the sides AB, BC   and AC

And call the center of the incircle  O

So....we can divide the larger triangle into three separate triangles  - AOB, BOC  and AOC

The inradius of the incircle forms an altitude of each triangle since it is perpendicular to each side

So.....the  area, A,  of the larger triangle can be expressed  as

A = (1/2)  [ altitude * ( sum of the bases ) ]  = 

(1/2) [ (1) * perimeter ]  =

 (1/2)  [ (1)(AB + BC + AC ) ]  =

(1/2) [ 1 * 12 ]  =

(1/2) [ 12]  =

6 units^2

The first one is incorrect. It's not right to use a semicolon to separate items in a series like that.

                      x^2 - x +3

x-4  |   x^3 - 5x^2 +7x - 12

           x^3 -4x^2

          ________________

                    -x^2 +7x -12

                   - x^2 +4x

________________________

                            3x - 12

                            3x - 12

_________________________

                                  0    remainder

Let's do some Algebra, first

Letting x  = 0......the y intercept  is  at     - 3

We  have

1x^3  + 2x^2  - 3

Note that, if we add the ntegers in red and get 0, then 1 is a root  (an x intercept)

So  ( x - 1)  is a factor

To find if we have any other real roots....perform some polynomial division

            x^2  + 3x  + 3

x - 1  [  x^3  + 2x^2  +  0x   -   3  ]

            x^3  -  1x^2

          ______________________

                      3x^2  + 0x

                      3x^2   -  3x

                      ________________

                                 3x  - 3

                                 3x  - 3

                                 ______

The remaining polynomial  is    x^2  + 3x   + 3

The discriminant  is     3^2 - 4 (1) (3)  =  -3

Since this is < 0......we have no more real roots

So.....

The y intercept  is - 3

And the x intercept is  1

Here's the graph : https://www.desmos.com/calculator/kfgpcs3g09

CPhill: Thank you so much, Sir.

Thank you so much :D

Better than I deserve! Enjoying a break today from a busy week. 0^>

If the polynomial (quadratic) will not factor, then use the Quadratic formula

\(x = {-b \pm \sqrt{b^2-4ac} \over 2a}\)

We will have to assume  2x^2+4x+7 = 0       SInce you are looking for roots

For this    a = 2 b = 4  c = 7

-4 +-sqrt(-40)    /   4     =     -1 +-  1/2 i sqrt (10)   

Let   BP  = x       and  CP  = 20 - x

In this situation, the altitude will be a "geometric mean"....such that

x            6

__  =   _____

6         20 - x

Cross-multiply

x (20 - x)  =  36

20x - x^2  = 36      rearrange as

x^2 - 20x + 36  =  0        factor

(x - 2) ( x - 18)  =  0

Set each factor to 0   and solve for  x  and  we get that  x  =  2   or x  = 18

So.....depending upon which  vertex is labeled as B  and which is labeled C.....

BP = 2  and CP  = 18    or

BP = 18  and CP  = 2

Either way.....the hypotenuse is divided into a ratio of 18 : 2

Lucky!

Same here we are supposed to get 3-6 inches of snow tonight...

Good....but cold   !!!   LOL!!!

Gooooood. You Synth?

The triangle on the "left"  has a base  =   3sqrt (3)

The triangle on the "right" has a base = 3 / sqrt (3)  =  3sqrt (3) / 3   =  sqrt (3)

So.....the area  =

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

3 [ 4sqrt (3) ]  / 2   =

3 [ 2 sqrt (3) ]  =

6sqrt (3)  units^2

We could solve this.....where N  =  the number of cartons that hold 12 eggs

2(11)  +  12N   > 100 

22 + 12N >  100  

subtract  22 from both sides

12N >  78      divide both sides by  12

N >  78/ 12

N  >   6.5

So.....we must have  7  cartons holding 12 eggs

So....the minimum number of eggs is

22 + 7(12)

22  + 84

106

1.   I woud do it like this:

            This week's homework assignment includes an essay, a group project and a digital slide presentation.

2.   I would do this one in one of two different ways.

             The waiter provided terrible service - and expected a tip.

             The waiter provided terrible service...and expected a tip.

However, I am NOT an English major !

Last one

Factor  x^2 + 4x + 3   =  (x + 3) ( x + 1)

Factor   x^2+ x  =   x ( x + 1)

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

So......just like the last one.....we really have

( x + 3) ( x + 1)              (2x - 1) ( x - 1)

____________      x     ______________

   2x   -  1                        x  ( x + 1)

"Cancel"   the  ( x + 1)   and  2x - 1  factors    and we have

( x + 3) ( x - 1)

___________

         x