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F(x) is just a substitute for y     it means that y is a function of x

f(x) = 3x+2   is the same as  y = 3x+2

Let x = slower cyclist    then faster is x + 2

   combined speed =   2x+2

      in 2.5 hours they cover 50 km

         rate * time = distance

           (2x+2) (2.5) = 50    solve for x

            5x + 5 = 50

             5x = 45

               x = 9  and the faster is  11 km/hr

This is the equation of a bowl shaped  parabola with vertex   1/2  , 5     which is the turning point      ..... NOT  (1,5)

(2x-1)^2 + 5     can be re-written

4(x-1/2)^2 + 5     

Too late; the correct answer is right above your post. 

By using the real numbers, we can simplify this simple question (and answer) even more.

Let a side of the square be 2 units

(I'll use hecticar's diagram)

1/    ∠a ≅ ∠c = arctan(DQ / AD) ≈ 26.565º

2/    ∠b = 90º - (∠a + ∠c) ≈ 36.87º

3/     sin(∠b) = 0.6    or   3/5 

1, 9, 36

Let SV be  x

x (x + 3) = 3 * 18

x² + 3x - 54 = 0       (solve with the quadratic formula)

x = 6

SW = 9

The discriminant must be a perfect square: \(a^2-24a=k^2\rightarrow a(a-24)=k^2\). From here it is casework: https://www.wolframalpha.com/input/?i=solve+a%28a-24%29%3Dk%5E2+over+the+integers

Is g' the derivative or inverse function?

These are quite annoying :(

We can let f(x)=ax^3+bx^2+cx+d

We get 27a+9b+3c+d=2, 64a+16b+4c+d=4, 125a+25b+5c+d=-3, 216a+36b+6c+d=-1. We want to find d. 

It is a giant bash, but we get d=-238

(z - 3)^2 + (z - 5)^2 = -8   expand and simplify to

2z²  -16z + 42 = 0   

Quadratic Formula shows z =   4 +- i sqrt 5

1 /4     /  8

1/4   /   8/1

=   1/4  *  1/8  = 1/32

There are   2* pi  Radians in a full circle

There are 360 degrees in a full circle

so   1   Radian = ~ 57.29578 ^o

aawww, you know what I did?????...I did not follow the order of operations when I checked the roots.....Thank you for your time, I do appreciate..

Look at the graph. (-4) and (6) are the ends of the line. This means that x is inbetween these numbers. To graph the inequality, we need to graph each one seperately. From (-4), we see that x > -4. From 6, we see that x < 6. It should be very simple from there. 

Hello there asinus, thanks for the reply, much appreciated! I see your "to be continued"answers for r squared and s squared, and I;ve been trying to finish it on my own. I just dont know how to add the values together, please help! Thank you

Hello there hectictar, thank you so much for the reply, I've been stuck on this 1 for a while and I cant show enough support, I just have 1 question. how do we know that sin B is equal to sin(90 degrees+2a)? Thank you so much

y = x^2 + 2x + 7  is a bowl shaped parabola.....the only ways a line can intersect it in ONE point is if the line is VERTICAL (this one has a slope of 3) ...or if the line is sloped , it must be tangent to the parabola   ( a third way would be for the line to be horizontal (slope = 0) and tangent at vertex)

When the line is tangent to the parabola, the parabola will have the same slope as the line ( 3)

find where the slope = 3

derivative of the parabola equation

   2x + 2       is the slope at each point x .....   eqaute to 3

2x+2 = 3

 x = 1/2    then   y = 8.25

so the line has the point   (1/2 , 8.25 )

plug these into the line equation

8.25 = 3(1/2) + b           b = 6.75

Try this:  3 + 4 + 5 + 6 + 7 =25.   3 x 4 x 5 x 6 x 7 =2,520

Update :     I looked at it a little closer:

x-2  = 3 (sqrt(x+2))   square both sides

x^2 - 4x + 4   = 9 (x+2)

x^2 -13x -14  = 0

Quadratic equation shows x = -1   and 14

14 - 3 sqrt (14+2) = 2

14 - 3 (sqrt16) = 2             ( look at your step where you get    11 sqrt (16)   <---- this is where you made your error)

14 - 12 = 2                           so 14 IS an answer     only -1 is NOT.

-1` - 3 sqrt(-1+2)    = 2  ?????    no it does not...it equals  -4

the solutions you obtained are both incorrect...you made a mistake somewhere (we ALL do !)

   ......go back and solve it again to get different answers

All of the angles of a triangle sum to 180^o

^{x + 2x + 6x = 180}

^{9x = 180}

^{x = 20}

I don't think I can solve this unless I assume the vertical lines ae parallel.....are they stated as being parallel?

This doesn't seem to be correct.