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(2x + 7) ( x - 5)  =   -43 + jx      simplify

2x^2 + 7x - 10x - 35 = -43 + jx

2x^2  - 3x - 35  =  -43 + jx

2x^2 - (3 + j)x + 8 = 0

This will have one real root when the discriminant = 0

So

(3 + j)^2 -   (4)(2)(8)  =  0

(3 + j)^2  =  64       take both roots

3 + j  =  8               or           3 + j = - 8

j = 5                      or           j = -11

P("choose fair coin") = 24/25

P("choose unfair coin") = 1/25

P("6H under the condition fair coin")\(= 2^{-6} = \frac{1}{64}\)

P("6H under the condition unfair coin")=1

let a=P("choose fair coin") * P("6H under the condition fair coin") = 24/25 * 1/64

let b=P("choose unfair coin") * P("6H under the condition unfair coin") = 1/25

\(\frac{b}{a+b} = \frac{m}{n} = \frac{8}{11}\)

m+n=19

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The easiest way to find the answer is to let x = 0

This will result in the fraction  -6/-8 =  3/4    which is the y intercept

Notice that the only graph that has this y intercept  is the one on  the top right

[ All the other graphs have y intercepts of 1 ]

A basement has a 24-foot by 32-foot rectangular floor. The basement is flooded with water to a depth of 18 inches. Three pumps are used to pump the water out of the basement. Each pump will pump 8 gallons of water per minute. If a cubic foot of water contains 7.5 gallons, how many minutes will it take to pump all of the water out of the basement using the three pumps?

Note that  18 in  =  1.5 ft

So....the cubic feet of water in the basement =   24 x 32 x 1.5   =  1152 ft^3

Each pump pumps out   8 gal  of water / minute  

So....the number of cubic feet they pump out each minute =  8 / 7.5  = 16/15 ft^3 min

So...three of them pump out  3 (16/15)   = 48/15 =  16/5   ft^3   min

So.....the number of minutes to pump out the basement is    1152 / (16/5)   =  1152 * 5 /  16 =   360 minutes =  6 hours 

1.) If, \(z = \frac{ \left\{ \sqrt{3} \right\}^2 - 2 \left\{ \sqrt{2} \right\}^2 }{ \left\{ \sqrt{3} \right\} - 2 \left\{ \sqrt{2} \right\} } \), find \(\lfloor z \rfloor\).

The fact that \(\lfloor \sqrt{2} \rfloor = \lfloor \sqrt{3} \rfloor = 1\), should help. Rewrite the roots in brackets with \(\{\sqrt{2}\} = \sqrt{2} - 1\) and \(\{\sqrt{3}\} = \sqrt{3} - 1\). 

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5 units left, 3 units up

This is the graph of the square root function shifted to the right by one unit  with a vertical stretch of 2

Thus....the correct graph is the one at the top right

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y = - 11.41x^2 + 154.42x - 143.9   ⇒   First answer

Another quick way is if you have a calculator/are allowed one, you can see which tan^(-1) corresponds to. For example, mine is yellow and above the "tan" function and so is my shift button. So if I click shift and then the tan button, I can input my value and it will be automatically calculated when I press "=." 

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There are 7 possibilities (A~G) for the first letter, 7 for the second, and 7 for the third. Multiplying this together, we get 7^3=343 total ways to make a set of 3-letter initials. 

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The graph is shifted to the left 6 units

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This is somewhat confusing.....I have to think about what I'm doing!!

The function is  called the "ceiling function"....meaning the least integer  greater than or equal to x

The easiest way to see this is to plot some values

x           ceiling x                ceiling x   - 3

-3            -3                           -6 

-2.5         -2                           -5

-2            -2                           -5

-1.5        -1                            -4

-1           -1                            -4

-0.5         0                            -3 

0             0                            -3   

0.5          1                            -2     

Note the the graph at the top left seems to be the correct one....     

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stop spamming please

\(\text{well the first thing I'd do is look at }-1 \leq x \leq 1\\ f(x) = |x-1| \text{ which decreases linearly from 2 to 0 on this interval}\\\text{That rules out the top left and bottom right graphs}\)

\(\text{Next I'd note that }f(8) = -2\\ \text{This rules out the bottom left graph}\\ \text{and thus the winner is the top right graph}\)

the left corner one should be the answer because...

f(x)=x^2-x-2

f(0) = 0^2-0-2 which is -2      plug numbers in

the bottom left is the only one with points (0,-2)

(CPhill's  is much more mathematical lol)