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x² + 3x - 28   =   (x - 4)(x + 7)   =  ( x + (-4) )( x - (-7) )   =   (x + a)(x - b)

x² - 10x - 56   =   (x - 14)(x + 4)   =   ( x + 2(-7) )( x + 4 )   =   (x + 2b)(x + c)

So...

a  =  -4  ,   b  =  -7  ,   c  =  4

And...

2c - a   =   2(4) - -4   =   8 - -4   =   12

(r + 3)  / ( r - 2)   =  (r - 1)  / ( r + 1)      cross-multiply

(r + 3) ( r + 1)  =  (r - 2) (r - 1)     simplify

r^2 + 4r  +  3   =   r^2 - 3r  +  2         subtract r^2 from both sides

4r + 3   =    -3r  +  2           add 3r to both sides, subtract 3 from both sides

7r  =  - 1       divide both sides by 7

r  =  - 1/7

For the slopes:

Use the slope formula:  m  =  (y₂ - y₁) / (x₂ - x₁)

Slope of AB:  [ (b + d) - b ] / [ (a + c) - a ]  =  d / c

Slope of DC: [ d - 0 ] / [ 2c - c ]  =  d / c

Since these slopes are equal, these sides are parallel.

Slope of AD:  =  ( b - 0 ) / ( a - c )  =  b / ( a - c )

Slope of BC  =  [ (b + d) - d ] / [ (a + c} - 2c ]  =  b / ( a - c )

Since these slopes are equal, these sides are parallel.

I actually think there is a more efficient way to do this........maybe someone else knows how

AB / CB   =  1/2  ⇒   2AB  = BC

AD/AB = 3   ⇒ 3AB  =  AD

BD/ AB  =  2  ⇒  2AB  = BD

So  this implies that  BD  =  BC

So....B  must  be the midpoint of  C and D  =   (2,2)

2AB  =  BC   ⇒  AB  =  BC/2

3AB  =  AD  ⇒  AB  =  AD/3

So logic dictates that A  must be the midpoint of BC  =  (1,3)

So....AB  =   sqrt [ (1^2 + 1^2) ]  =  sqrt (2)

And  AD  =  sqrt (3^2  + 3^2)  = sqrt (18)   =  3sqrt (2)  =  3AB

Proof 

AB   =  sqrt (2)

CB  =  sqrt [ 2^2  + (4 -2)^2]  =  sqrt (4 + 4)  =    sqrt (8)  = 2sqrt (2)

So  AB /CB  =  sqrt (2)/ 2sqrt (2)  =  1/2

And

AD  =  3sqrt(2)

BA  =  sqrt(2)

So  DA /BA  =   3sqrt(2)/ sqrt (3)  = 3

And

DB  = sqrt [ ( 4 - 2)^2  + 2^2 ]  =  sqrt (2^2 + 2^2 )  =  sqrt (8)  = 2sqrt(2)

BA  =  sqrt(2)

So DB / BA  =   2sqrt(2)/ sqrt (2)  =  2 

So  A  =  (1 ,3)

And   2x + y   =   2(1) + 3     =      5

Wow thanks!!

We have that

a*4^2 + 5  = a*6^2  + 7   simplify

16a + 5  =  36a + 7  rearrange as

-2  =  20a   divide both sides by  20

-2/20  =  a   =  - 1/10

Not totally sure about this...but.....here's my best attempt...

abs [ 2n^2  + 23n   + 11 ]   factor

abs(2n  + 1) * abs(n + 11)

Note that this will  be a possible prime  if either factor  =  ±1 

But 2n + 1  will = 1  only when  n  = 0.....and the other factor will = abs ( 11 )  = 11

So....when  n = 0, the result will be prime, i.e, 11

And 2n + 1  will equal  - 1  when  n  = -1.....and the other factor will =  abs(-1  + 11) = 10......but this isn't prime

And n + 11  will equal  1  when n  =  -10

And the other factor will  be 2(-10) + 1  =  -19   which is prime for abs (2n + 1) = abs (2*-10 + 1) =

abs(-19)  =  19

And  n + 11  will =  - 1  when n  =  -12  ....so abs (-12 + 11)  = abs(-1)  =  1

And the other factor will be  abs (2(-12) + 1)  =  abs (-23)  =  23

So.....this will be prime  when n   = -12

So.....the  integers producing prime results  for  abs [ 2n^2  + 23n   + 11 ]  are 

n = 0 , n  = -10 and n = -12

EDITED ANSWER.....still don't know if it's correct, or not....!!!!!!

Hours * Rate per hour  = Total Amt

So

(t + 1) (3t - 3) =  (3t  - 5)(t + 2)  + 2    simplify

3t^2 + 3t  - 3t - 3  =  3t^2  - 5t + 6t - 10  + 2

-3   =  1t - 8         add 8  to both sides

5  =  1t

5  = t  

Check

(5 + 1)(3*5 - 3)  =  (3(5) - 5 ) ( 5 + 2)  + 2   ???

(6)(12)  = (10)(7)  + 2

72   =  70  + 2

So.....t  = 5  is correct

Thanks, CPhill and azsun!

25.  We have this equation

y  =  497 (1/2)^x/38    where  y is the amount remaining after x  days have elapsed

[ This is the first answer ]

After 4 days have elapsed, the amt remaining is

497(1/2)^4/38  ≈  462.06  kg

26.    log ₄ 1024

We can use the change-of-base theorem, here

The answer is given by :

log 1024  /  log 4   =   

5

To start, we can find the -intercept of each of these lines. Using this, we can calculate the length of that side of the triangle, and use it as a base. Letting x=0 in the first equation gives y=-2 as a y-intercept. Letting x=0 in the second equation gives \(3y=12\Rightarrow y=4\) as a y-intercept. Therefore, the triangle has a length of \(4-(-2)=6\) on the  y-axis.

The height of the triangle will be equal to the -coordinate of the intersection of the two lines. So, we need to solve for x in the system: \(\begin{align*} y-3x&=-2\\ 3y+x&=12 \end{align*}\) Multiply the first equation by 3, then subtract the second equation as shown: -10x=-18Therefore, \(x=\frac{18}{10}=\frac{9}{5}\) . This is equal to the height of the triangle. The area will be \(\frac{1}{2}\cdot \frac{9}{5}\cdot 6=\boxed{\frac{27}{5}}\)

We need to find two-thirds of the difference in the x values of points R and S and add this to the X value of point R . Then we will find two-thirds of the difference in the y values and add this to the y value of point R . When we solve for the x value we find that \(\frac{2}{3}\times (15-(-6))=\frac{2}{3} \times 21 = 14\) , and -6+14=8 , so x=8 . When we solve for y we find that \(\frac{2}{3} \times (-5 -19) = \frac{2}{3} \times (-24) = -16 \) , and \(19 + (-16) = 3\) , so y=3 . The coordinates are \(\boxed{(8,3)}\) .

Look  at the graph here:

https://www.desmos.com/calculator/tleqe1iopp

We have a triangle

Let the base be formed by the segment  connecting  (0,-2)  and (0, 4)

And the length of this base  =  6  units

Now.....an altitude can be drawn from the point  (1.8, 3.4)  to the y axis

And this altitude is  1.8 units

So.....the area of this triangle  is

(1/2)(base)(altitude)  =

(1/2)(6)(1.8)  =

.9 * 6  =

5.4 units^2

substituting 5 in the question doesn't give 2 times the dollars

Thanks :)

Alright thanks

I got the same answer on a rough page but looking at the answer I was like 'w*f is this'

If f(x) = (x+1)(x^2+1)(x^4+1)(x^8+1)(x^16+1)(x^32​+1), then in simplified form f(2) = 2n-1. Determine the value of n.

f(2)  = 

(2 + 1) (2^2 + 1) (2^4 + 1)(2^8 + 1) (2^16 + 1)(2^32 + 1)  =

(3)(5)(17)(257)(65537)(4294967297)  =

18446744073709551615  = 2^n  - 1       add 1 to both sides 

18446744073709551616  =  2^n        take the log of both sides        

log (18446744073709551615)  =  log (2)^n     and we can write

log (18446744073709551615)  = n* log (2)    divide both sides by log 2

log (18446744073709551615) / log (2)  =  n  =  64 

I think if I answer any more math questions my mind will b**w up cuz its mixed with chemistry and maths and that a very bad mixture

nope, t=5

25 is a messed up question but for 26 the answer is 5

by 'WTH' i thought u were trolling but there was actually a question, lemme have a look at it.

nevermind I got it.

\((t+1)(3t-3)=(3t-5)(t+2)\)

Expanding gives

\(3t^2-3t+3t-3= 3t^2+6t-5t-10\)

Get the like terms together

\(3t^2-3t^2 -3t+3t +5t - 6t=-10+3\)

3t² cuts out and the remaining equals to -t but u are looking for a positive t so so multiply both sides by (-).

therefore this gives

\(t=10-3\)

\(t=7\)

So on Tuesday, u worked 8 hours and got 18$ and Andrew worked 16 hours and got 9 dollars. And also the question should be

'At the end of the day, I had earned two times more than he had. What is the value of t?'

For number 26, I'm getting (A), 5....