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2x + 7  =  -5                        -x^2  - x  +  2  = -5

2x  =  -5 - 7                          x^2  + x  - 2  =  5

2x  = -12                              x^2  + x    =   7         complete  the  square on x

x = -6                                    x^2  + x + 1/4  = 7 + 1/4

                                             (x + 1/2)^2  =  29/4         take  the positive root

                                               x + 1/2  =   sqrt (29)/2

                                               x =  (1 + sqrt (29) )   /2

Sum =    -6   +   (1 + sqrt 29) / 2  =    (-11 + sqrt 29)   / 2

If \(\frac{9x^3\:+\:4x^2\:+\:11x\:+\:7}{x^2\:+\:bx\:+\:18\:}\) is real, its denominator must never equal zero

The discriminant of x^2+bx+18 is b^2-72

if x^2+bx+18 doesn't equal zero, than: 

b^2-72<0

b=+/-(6 sqrt (2))

so the greatest integer is 8

JP

Nice work, heureka, as always!!!

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QS = TU = 2

QT = QR = 8

QW = 5

RV = 4

TW = sqrt(QT² - QW²)

RT = sqrt(TW² + RW²)

RW / RT = RV / PR

PQ = PR = (RT * RV) / RW

No such consecutive postivie odds   exist

Let  the  integers  be       2n - 1  ,  2n + 1 and 2n + 3

So

(2n + 1)^2  - (2n - 1)^2  =  348

4n^2  + 4n + 1  - 4n^2 + 4n - 1  = 348

8n =  348

n = 348/8   =  87/2

2 (87/2)  - 1 =      86

2n + 1  =  88

2n + 3  =  90

Note

88^2  - 86^2   =  348

And

90^2  -88^2  >   0

thanks for the correction

A =  16

B = 96

96 - 16   =   80

Smallest multiple of 8: 16

Largest multiple of 24: 96

96 - 16 = 80

(Didn't read the question before this edit, just realized it has to be 2 digit numbers) 

Oops!!!....see  my edit above   !!!!

Let's assume A is the smallest. Then b would be A + 2, and C would be A + 4. Sub those values in. Can you take it from here? 

(I also hope i'm correct)

no i think the equation is wrong i didnt pay attention to the graph nor the form, soz 

< Find all integers n such that when the clock strikes n o'clock, the acute angle between the hour and minute hands is 90 degrees. Enter your answer as a comma-separated list. >     

Note that 90^o is not an acute angle, but never mind that.  When the clock strikes the hour, that means the minute hand is on the 12.  For the angle to be 90^o the hour hand must be on either the 3 or the 9.  Assuming a 12-hour clock.  

_.

Possible rolls

1 + 1    =  2

4 + 4  =   8

We  have  36 possible outcomes

P(2)  =    1/36

P(8)  =    5/36

Total Probability =   1/36  + 5/36     =  6/36   =   1/6

wrong...

See  the  graph  here  :  https://www.desmos.com/calculator/qzfidyzqfw

We  have  two  triangles  whose  base and  height   =  6....so...the  area =   6 * 6    = 36

By Pick's Theorem

Area =   Number of interior lattice points  +  Number  of lattice points on the  boundary/2   - 1

The  number of lattice points on the  boundary  =    1 + 6  + 6 +  11  =   24

So

36 =   I  +  24 /2    -  1

36  = I  +  12 -  1

36 = I  + 11

I =  25  =  25 interior lattice points

If  6 is replaced  by 100.....the  area =   100^2 =  10000

The  number of lattice points on the  boundary =  1 + 100 + 100 + 199 = 400

By Pick's Theorem

10000  =  I   +  400/2  - 1

10000 = I  + 200 - 1

10000 =  I  +  199

I  =   9801   lattice points in the  interior

EDIT  TO CORRECT A  PREVIOUS ERROR   !!!!

Time spent on  plane  surface =   8km /  80km/h =  (1/10) h

Distance / rate   =  time

Let  x  be  the downhill part  and y  be  the uphill part  from the  city to the  beach

Then...on the return trip, the roles   are reversed.....y is the downhill part and x the uphill part

So   we  have  this system  of total times  ( in hrs)

x  / 100     + y/ 60  +   1/10  =    2

y/100  + x / 60   +  1/10   = 3           simplify

x/100 + y/60  =  1.9

y/100  + x/60  =  2.9               mutiply both  equations through  by  300.....arrange  as:

3x  +  5y  =  570     ⇒  mult through  by 5 ⇒      15x  + 25 y  =  2850

5x  +  3y   =  870    ⇒  mult through by -3 ⇒     -15x  - 9y  =  -2610               add these

16y  =  240

y =  15     (km)

3x  +  5(15)  =  570

3x  +  75  =  570

3x = 495

x = 165  (km)

Total  distance =    165  +  15 +  8  =  188 km

You don't specify  a leftmost  bound but if  we  take it  as  the   line  x  =1, we have  9 lattice points

See  here  :  https://www.desmos.com/calculator/oo5vppcbet

thanks :D, have a great day

Thx, Viola....you, too  !!!!

I  know  a  few

a.    t =   200  +  3h

b .       Since  we  don't  know  the  flat fee, the   total  cost  is   t  = 6h  

c.     We  can solve this

200  +  3h  <  6h

200 < 3h

h  >   200 / 3     =   about  67  hours

d.   Average  cost  per  hour  (at  the tennis club)  =    (200 + 3h)  / h

Here's  the graph  (I've substituted  x  for  h, but it's the same  graph )

https://www.desmos.com/calculator/ojdu2a77xa

I see. Thanks to you and cPhill. I hope you all have a nice day!

What's  the  equation for  the  curve   ???

If we   have one real solution  then we  have a double root......which means that the  discriminant = 0

So

m^2  - 4(m + 2n)   =  0

m^2  - 4m  -  8n  =  0

Maybe  many solutions for  this but  one possible solution  is   m = 8   and  n = 4

So  we  have

x^2   +  8x   +   16 =  0

(x + 4) ^2   = 0

x  = -4   is  the  only solution

Just  a slight  mistake  by JP....

We  have   a right triangle   with   legs of  4  and 13

The  radius^2   =   4^2  + 13^2   =    185

So.....the  area  =   185 pi

It is right; it just isn't in ax+by+c > or >= 0 form.

There is no constant term.....the  expansion is

x^8 + 4 x^5 + 1/x^4 + 6 x^2 + 4/x