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For a round trip the uphill = downhill distances   and the flat = 16 km 

   so   U = D        Total roundtrip distance = U + U + F                                       U = uphill   D = downhill

U/100 + U/50   + 16 / 80 = 5 hours

(U + 2U) / 100 = 4.8

results in

U =160     and   F = 16    and  D = 160     ( roundtrip )

     total ONE-WAY distance =  (  160+160+ 16 )/ 2 = 168 km

Nice work  guest !

\(x^2+a^2-2ax + 4x^2+b^2-4bx= x^2 + 9 - 6x + 4x^2\)

Now simply the left side and simplify the right side independently of each other.

\(x^2+a^2-2ax + 4x^2+b^2-4bx= x^2 + 9 - 6x + 4x^2\\ 5x^2+(-2a-4b)x+(a^2+b^2)=5x^2-6x+9\)

Now equate coefficients

\(5=5\\~\\ -2a-4b=-6\\ a+2b=3\\ a=3-2b \\~\\ (3-2b)^2+b^2=9\\ 9+4b^2-12b+b^2=9\\ 5b^2-12b=0\\ b(5b-12)=0\\ b=0\quad or \quad b=12/5 = 2.4\\~\\ if\; b=0, \\\quad a=3-0=3\\ if\;b=2.4, \\\quad a=3-2*2.4=3-4.8=-1.8\\ \text{so I have }(0,3)\quad and \quad (2.4,-1.8) \)

You need to check these by plugging them into the ORIGINAL equation.

_+-5=4             add + 5 from both sides of the equation....

a - b = 2 (c+d)          but b = a-2    sub that in

a - ( a-2)  = 2 ( c+d)

2 = 2 ( c+d)

1 = c+d                 now   d = c+8     sub that in

1 = c   + c+8

1 = 2c + 8

-7 = 2c

c = - 7/2

_ - 5 = -9     add +5 to both sides

_  = - 4

You will end up with more complicated polynomials, and the person who asked the question may not be able to fully expand it out or solve it with a cubic equation.

Though if you do factor it out it may be a better strategy if you know how... :)

So a^3 + b^3 = (a + b)(a^2 - ab + b^2). 

Substituting in, we have 812 = 14(a^2 - ab + b^2).

Simplifying we have 58 = a^2 - ab + b^2.

a^2 - ab + b^2 = (a + b)(a + b) - 3ab. 

Substituting in, we have 58 = (14)(14) - 3ab.

Now we have ab = 46.

Then we also have a^2 - ab + b^2 = (a - b)(a - b) + ab.

Thus, 58 = (a - b)^2 + 46.

(a - b)^2 = 12.

\(a - b = \pm{2\sqrt{3}}\)

Now we can sove for a. If we add the two equations together, we get 2a = 14 + 2sqrt(3)

Thus, \(a = 7 + \sqrt{3}\), and \(a = 7 - \sqrt{3}\).

Substituting in, we have \(b = 7 - \sqrt{3}\), and \(b = 7 - 3\sqrt{3}\).

Thus, our possible for solutions \((a, b)\)are:

\((7 + \sqrt{3}, 7 - \sqrt{3})\)

\((7 - \sqrt{3}, 7 - 3\sqrt{3})\)

x = 8-y

(8-y)^2 + y^2 = 38

y^2 + 64 + y^2 - 16y = 38

2y^2 - 16y + 26 = 0

y^2 - 8y + 13 = 0                 y = \( {8 \pm \sqrt{64-52} \over 2a} = {8 \pm \sqrt{12} \over 2a}\) = (8+-2√3)/2 , which is 4+-√3. 

(x, y) = (4-√3, 4+√3), (4+√3, 4-√3)

from equation 1 we get b = 14-a

substituting that into the second equation we get a^3+(14-a)^3=812

now you can try to expand it out, if you need help just ask

I get \(x^2+a^2-2ax + 4x^2+b^2-4bx= x^2 + 9 - 6x + 4x^2\)

\((x-a)^2 + (2x-b)^2 = (x-3)^2 + (2x)^2\)

I'd start by expanding it out.  Have you done that?  Show us.

Cphill is correct....    I AGAIN missed the tiny ' - ' sign !

Use a calculator to find    arc cos (.111) = ~ 84 degrees

192 , 301 , 300 , 300 , 300 , 300 , 300 , 300 , 300 , 300 Total with repeats = 2893
(181, 272,  271,   271,    271,   271,    271,   271,    271,    271)Total  w/o  repeats = 2621

Count of all digits between 1 and 1,000 inclusive (with repeats + without repeats).

q = quarters, d = dimes, n = nickels, p = pennies

Only 1 quarter:

1q + 2d

1q + 1d + 2n

1q + 1d + 1n + 5p

1q + 1d + 10p

1q + 4n

1q + 3n + 5p

1q + 2n + 10p

1q + 1n + 15p

1q + 20p

9 ways

No quarter:

4d + 1n

4d + 5p

3d + 3n

3d + 2n + 5p

3d + 1n + 10p

3d + 15p

2d + 5n

2d + 4n + 5p

2d + 3n + 10p

2d + 2n + 15p

2d + 1n + 20p

2d + 25p

1d + 7n

1d + 6n + 5p

1d + 5n + 10p

1d + 4n + 15p

1d + 3n + 20p

1d + 2n + 25p

1d + 1n + 30p

1d + 35p

9n

8n + 5p

7n + 10p

6n + 15p

5n + 20p

4n + 25p

3n + 30p

2n + 35p

1n + 40p

45p

30 ways

30 + 9 = 39 ways

(x^2  + 7/x)^4

There is no constant term

Note the expansion :

x^8 + 28 x^5 + 2401/x^4 + 294 x^2 + 1372/x

x^2 - l x^2 l

Note that the absolute value of  x^2  =  x^2

So......this always evaluates to   0

A slight mistake above

We know  that c = 4   and a  =1

So

-10  =  (1)^2  + b(1) + 4 

- 10  = 5 + b

-10 -5  = b   =  -  15

The form is

f(x)   =x^2  - 15x  +  4

We need to find the area of the table top

The radius of the table top can  be found as

4.71 / ( 2*3.14) =  .75 ft

The area =  3.14 (.75)^2   = 1.776 sq ft  = 1.8  sq ft

a/b  *   b/c  = a/c

sqrt10 / sqrt 21     *  sqrt 14/sqrt15   =

sqrt 140  / sqrt 315

2 sqrt 35  / 3 sqrt 35   = 2/3

1.8 square feet of plastic.

9/1000

one.