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We have the equation \({A \over (x - 1)} + {B \over (x + 1)} = {(x + 3) \over (x^2 - 1)}\)

Multiply by \(x^2 -1\): \(A(x+1) + B(x-1) = x + 3\)

From this, we know that \(A + B = 1\) and \(A - B = 3\), so \(A - B = \color{brown}\boxed{3}\)

See the graph here

https://www.desmos.com/calculator/nztbhkeg63

The total area  is a square with a side of 1

We have a right triangle  with  legs of  (1/4) and (1) which is  not  included in the inequality region

Its area  =  (1/2)(1/4) (1)  =  1/8

So....the probability  that   y ≤ 4x  is

[ 1 - 1/8)] / 1  =   7 / 8

q(-1)=3

No worries - we all make them!

BuilderBoi  already found the area to be 25/9

So the side of the square =  sqrt (25/9)  = 5/3

Here's the proof

The  top of the  square  has the length   5/9 + 5/9 + 5/9 =  15/9 =  5/3

By symmetry  the   top edge of one of the two rectangles at the bottom = (5/3) (1/2) =  5/6

And the area of one of these rectangles =  5/9

So  the other dimension  of one of these rectangles must be  (5/9) / (5/6) = 6/9 =  2/3

And AB =  1

So....the  side of the  square is also   1 + 2/3   =  5/3

See https://web2.0calc.com/questions/geometry_363

Already answered by BuilderBoi, further down in the thread.

Nice one, there, BuilderBoi...

4- 8i

______      multiply numerator/ denominator by   4 - 8i

4 + 8i

(4 -8i)^2

_________

16 - 64i^2

16 - 64i + 64i^2

____________

        80

-48 - 64 i

_______

    80

- 3  - 4i

_____

     5

  (3 + 4i)

________

     -  5

Very nice, BuilderBoi !!!

Let one of the sides of the rectangle containing AB be x. 

The area of this rectangle is \(x \times 1 = x\)

There are 5 of these, so the area is \(5 \times x = 5x\)
 

But, note that because 3 rectangles share side AB, their width is also x, meaning 1 side of the square is \(3x\).

So, we have the equation \((3x)^2 = 5x\). 

Solving, we find \(x = {5 \over 9}\), meaning the area is \(5 \times {5 \over 9} = \color{brown}\boxed{25 \over 9}\)

Add all the equations and we get

10a + 10b + 10c + 10d =  0

So   dividing through by 10

a + b + c + d  =  0

This is more like how it should look. Sorry, guys.

Answer: \(4\sqrt{1486}\over{-49}\)

Solution:

h(t) should be equal to 4.

So:

-4.9t² + 18t - 0.4 = 4

-4.9t² + 18t - 4.4 = 0

Putting this into the quadratic formula gives:

\(-18\pm\sqrt{237.76}\over{-9.8}\)

You need the bigger one of these roots minus the smaller one, which is equal to 2 times the square root of 237.76 over -9.8. This is also equal to \(4\sqrt{1486}\over{-49}\).

Multiply through by x^2

6x^2  = 35x  - 49  + 5      

6x^2  = 35x - 44      rearrange as

6x^2 - 35x + 44  =  0

(x - 4) ( 6x - 11)  =  0

x - 4 = 0                 6x - 11  =  0

x  = 4                       6x  = 11

                                 x  =11/ 6

x =  11/6  ,  4

Let's look at the times when it is at 4m

-4.9t^2  + 18t  - .4  =  4 

-4.9t^2 + 18t  - .4 - 4  = 0

-4.9t^2 + 18t - 4.4  = 0          multiply through by -10

49t^2  - 180t  + 44  = 0

Using the Quadratic Formula

             180  ± sqrt [ 180^2 - 4(49) (44) ]             180  ± sqrt [ 23776]

t =        ____________________________ =    _________________

                              2 (49)                                                    98

The time when it is above   4m  is given by

sqrt [ 23776] / 98    -  - sqrt [ 23776] / 98  =

2sqrt [ 23776 ] / 98  = 

sqrt [ 23776]  /  49       sec

\(In\;the\; \Delta ABC, we\; have\; the \;\angle A\;=90^{\circ}.\;sin(B)\;=\; \frac{5}{7}.\;Find\;cos(B).\)

Sorry guys, I kinda f'ed up cos and sin in the drawing, but the calculation is correct.

There are 6 possible outcomes for each dice, so in every throw You have 6 x 6 x 6 = 216 possible outcomes.

There is only one way the sum of three dice can become 18, that is if they all turn up as 6.

So the probability of getting 18 as a sum in just one throw is \(\frac{1}{216} = 0,00462962962962963 \approx 0.0046\)

For the function to have a domain of all real numbers, the denominator must ave imaginary solutions. 

For an equation to have imaginary solutions, the discriminant(b^2 - 4ac) must be negative. 

So, we have the equation \(1 - 4c < 0\). The smallest possible integer that works is \(\color{brown}\boxed{1}\)

Start with numbers ending in 5.

The prime restriction is already satisfied, so we don't worry about that.

Then, there are 8 choices for the hundreds digit (anything but 0 or 5) and 8 for the tens digit (anything but the hundreds digit and 5), which makes for \(8 \times 8 = 64\)

Now, if the final digit is 0, there are 9 choices for the hundreds (anything but 0), and 8 for the tens (anything but the hundreds and 0), which makes for \(9 \times 8 = 72\)

But, we need to account for numbers that have no primes. The digits that aren't prime are (0, 1, 4, 6, 8, and 9). 

There are 5 choices for the hundreds and 4 for the tens, which makes for \(5 \times 4 = 20\) numbers that don't work. 

So, there are \(64 + 72 - 20 = \color{brown}\boxed{116}\) numbers that work. 

I hope You can take it from there.

Make a table and fill it in with variables.  

                blue eyes        brown eyes       green eyes

7th-grade       9                        a                        b

8th-grade        c                        9                       d

We have the following equations, and we need to solve for b: 

\(9 = b + d \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ (i)\)

\(9 + a + b = 3(9 + c) \ \ \ \ \ \ \ \ \ \ \ (ii)\)

\(9 + a = 2(9 + c + d) \ \ \ \ \ \ \ \ \ \ \ (iii)\)

\(18 + a + b + c + d = 46 \ \ \ \ \ \ (iv)\)

After solving for b, we find \(b = \color{brown}\boxed{7}\)

Great, guest, and the sum is?

The length from the orign to A(8,0) is 8. The formula for the area of a trangle is Area = 1/2 x a x b.

Let the length from the orign to (8,0) be a. To find b we can set 16 = 1/2 x 8 x b, or 16 = 4 x b. So b = 16/4 = 4.

So the length from the orign, along the y-axis to B must be 4, giving us the coordinate B(0,4).

To calculate the slope of a line from A to B we subtract the coordinates of the y-components and divide these by

the diffrence between the coordinates of the x-components, so slope = \(\frac{B_y - A_y}{B_x-A_x} = \frac{4 - 0}{0-8} = \frac{4}{-8} = -\frac{1}{2} = -0.5\)

\(f(n) = n^2 + n + 17\)

\(f(10) = 10^2 + 10 + 17 = 100 + 10 + 17 = 127\)

\(f(9) = 9^2 + 9 + 17 = 81 + 9 + 17 = 107\)

\(f(10) - f(9) = 127 - 107 = 20\)