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There appears to be a maximum value for d which is = 57

a     b      c   d

31, 40, 44, 57

The length of the segment between the points (2a, a-4) and (4, -1) is 2*sqrt10 units. What is the product of all possible values for a?

Sorry...made a basic mistake in my answer,,,,corrected answer above       

The base of the pyramid is a square, so its area is found by:  Area = s²  =  12²

Each of the sides is a triangle, so its area is found by:  Area  =  ½·b·h  =  ½·12·15

(the slant height of the pyramid is the height of each of its sides)

To find the total area, add the area of the base to the area of its 4 sides.

Original problem:   3(x¹⁰ - x⁷ + 2x³ - x + 7) + 4(x³ - 2x² - 5)

Simplifying:            3x¹⁰ - 3x⁷ + 6x³ -3x + 21 + 4x³ - 8x² - 15

                               3x¹⁰ - 3x⁷ + 10x³ - 8x² - 3x + 6

The coefficients are 3, -3, 10, -8, -3 and, maybe 6.

At one time the constant term (the 6) was called a coefficient because it could be written as 6x⁰; 

but, now, it isn't always called a coefficient.

What did your teacher say?

The formula is:  y  =  5x - 274

The value of y is -69

Substituing:                        -69  =  5x - 274

Adding 274 to both sides:  205  =  5x

Divide both sides by 5:         41  =  x

The input is the x-value.

To find the length of the diagonals, use the distance formula:     distance  =  sqrt[ (x₂ - x₁)² + (y₂ - y₁)² ]

From point (1,0) to (7,8):      distance  =  sqrt[ (7 - 1)² + (8 - 0)² ]

From point (0,7) to (12,-2);  distance  =  sqrt[ (12 - 0)² + (-2 - 7)² ]

Find the values of teh two distance and place these values into your formula.

3)  Let the coordinates of point P be (x, y).

Using the distance formula:

     PA  =  sqrt[ (x - 4)² + (y - -1)² ]     --->    PA  =  sqrt[ (x - 4)² + (y + 1)²​ ]     --->     PA²  =  (x - 4)² + (y + 1)²  

     PB  =  sqrt[ (x - 6)² + (y - 2)² ]     --->    PB  =  sqrt[ (x - 6)² + (y +- 2)²​ ]     --->     PB²  =  (x - 6)² + (y - 2)²  

     PC  =  sqrt[ (x - -1)2 + (y - 2)2 ]     --->   PC  =  sqrt[ (x + 1)2 + (y - 2)2​ ]     --->     PC2  =  (x + 1)2 + (y - 2)2  

Combining these:

     PA2 + PB2 + PC2  =  (x - 4)2 + (y + 1)2  + (x - 6)2 + (y - 2)2  +  (x + 1)2 + (y - 2)2  

                                   =  (x² - 8x + 16) + (y² + 2y + 1) + (x² - 12x + 36) + (y² - 4y + 4) + (x² + 2x + 1) + (y² - 4y + 4)

                                   =  3x² - 18x + 3y² - 6y + 62 

                                   =  (3x² - 18x    ) + 3y² - 6y   )  +  62

                                   =  3(x² - 6x      ) + 3(y² - 2y  )  + 62

                                   =  3(x² - 6x + 9) + 3(y² - 2y + 1) + 32

                                   =  3(x - 3)² + 3(y - 1)² + 32

Let the coordinates of X be (a,b)   --->   PX  =  sqrt[ (x - a)² + (y- b)² ]     --->     PX²  =  (x - a)² + (y- b)² 

     3PX² + k  =  3[ (x - a)² + (y- b)² ] + k  =  3(x - a)² + 3(y - b)² + k

Setting these two equal to each other:  3(x - 3)² + 3(y - 1)² + 32  =  3(x - a)² + 3(y - b)²​ + k

tells me that a = 3, b = 1, and k = 32

thanks !! but i entered 2 and 3 as a list, and it marked me wrong. do you know what's up?

i think some parts of the question are missing. can you write the question again pls? :)

*****    edited   ****    edited *****

Neil    found   1 and 6      this means  (x-1)(x-6)         x^2 - 7x +  6

Chris  found   -1 and -4    this means  (x+1)(x+4)   =  x^2 + 5x + 6   

If the correct values are in red     then    the equation must be    x^2 + 5x +   6        which factors to (x+3)(x+2)    roots  -3 and - 2

ohh! thank you EP:))

Nevermind, got it!

The number to place in the blank is    '  121  ' 

You're welcome !!        I HOPE you understand it !

hm.. neither of those answers are right.

their combined speed is  80 ft /sec

Let's stop one train and say the other is travelling the 80/ft/sec

When their noses touch.....the moving train moves another 15o ft.....then the trains are exactly BESIDE each other

   the moving train needs to move ANOTHER 150 ft to completely pass the other....a total of 300 ft at 80 ft/sec

    300 ft  /   (80 ft/sec)  = 3.75 seconds to completely pass

thank you for breaking that down that really helped me understand it... I have a lot of trouble with this stuff.. I really appreciate it.

For payments made at the BEGINNING OF THE MONTH   (payment due annuity)

    ​P=PMT×((1+r)ⁿ−1)/r  ​×  (1+r)​

For payments made at the END OF THE MONTH (ordinary annuity....more common)

   P =PMT X ((1+r)ⁿ  -1)/r       I'll use this one

        PMT = 95      r = 3%/12months = .0025 % per month      n = 10 years * 12 months/year  = 120 months

 P = 95 ( ( 1+.0025)¹²⁰ -1 ) / .0025   =  $ 13 275.43

Would you mind solving that so I can confirm my calculations? I feel like I got an unrealistic number :

sqrt[(42sqrt3)/5]

this doesn't help... I don't under stand

2)                                   x² + y²  =  4x + 4y

                         x² + y² - 4x - 4y  =  0

       (x² - 4x       ) + (y² - 4y      )  =  0

        (x² - 4x + 4) + (y² - 4y + 4)  =  8                                (complete the square)

                       (x - 2)² + (y - 2)²  =  ( 2sqrt(2) }² 

The largest value of x will occur when the y-term becomes zero; so, let y = 2:

                      (x - 2)² + (2- 2)²  =  ( 2sqrt(2) }² 

                          (x - 2)² + (0)²  =  ( 2sqrt(2) }² 

                                    (x - 2)²  =  ( 2sqrt(2) }² 

                                        x - 2  =  2sqrt(2)

                                             x  =  2 + 2sqrt(2) 

This is 'completing the square '  exercise'

take 1/2 of the 'x' coefficient     then square it    11^2 = 121

x^2 + 22x + 121    =    (x+11)²