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How would I solve this equation algebraically?
\(\large{ \sqrt{5x^2+11}} = x + 5\)

\(\begin{array}{|rcll|} \hline \sqrt{5x^2+11} &=& x + 5 \quad & | \quad \text{square both sides} \\ 5x^2+11 &=& (x + 5)^2 \\ 5x^2+11 &=& x^2+10x+25 \\ 4x^2-10x-14 &=& 0 \quad & | \quad : 2 \\ 2x^2-5x-7 &=& 0 \\ \\ x &=& \dfrac{ 5\pm\sqrt{25-4\cdot 2 \cdot (-7) } } {2\cdot 2 } \\\\ x &=& \dfrac{ 5\pm\sqrt{25+56 } } { 4 } \\\\ x &=& \dfrac{ 5\pm\sqrt{81} } { 4 } \\\\ x &=& \dfrac{ 5\pm 9 } { 4 } \\\\ x_1 &=& \dfrac{ 5 + 9 } { 4 } \\\\ x_1 &=& \dfrac{ 14 } { 4 } \\\\ \mathbf{ x_1 } &\mathbf{ =}& \mathbf{ \dfrac{ 7 } { 2 }} \\\\ x_2 &=& \dfrac{ 5 - 9 } { 4 } \\\\ x_2 &=& -\dfrac{ 4 } { 4 } \\\\ \mathbf{ x_2} &\mathbf{ =}& \mathbf{ -1} \\ \hline \end{array}\)

\(x^2 + 24 = 0\\ x^2 = -24\\ x = \pm i\sqrt{24} = \pm 2i\sqrt{6}\\ x = 2i\sqrt{6} \text{ or }x= -2i\sqrt{6}\)

\(\text{the key is that } \dfrac a b \div \dfrac c d = \dfrac a b \times \dfrac d c\\ -\dfrac{21}{10}\div \dfrac{56}{24} = -\dfrac{21}{10}\times \dfrac{24}{56} = -\dfrac{9}{10}\)

well they've highlighted two points on this line so I guess they want you to use them.

\(\text{the two points are }(3,21) \text{ and }(6,12) \\ \text{we calculate the slope as }m = \dfrac{y_2-y_1}{x_2-x_1} \\ m = \dfrac{12 - 21}{6-3} = \dfrac{-9}{3}=-3 \\ \text{using the first point, point slope form is thus}\\ (y-21)=-3(x-3) \\ \text{and with a bit of rearranging we get }\\ y = -3x + 30 \text{ which is slope intercept form}\)

for (b) just plug in x=0 for the y intercept

we already found the slope as m=-3

so his delivery time decreased 3 minutes per day

1 Liter = 10^15 cubic Micrometers.

10^15 / 100 =10^13 cells in 1 Liter.

I'm seeing (5,20), (6, 12), (8,8) as the possible cool rectangles so 

100 + 72 + 64 = 236

Well,

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X == 6 and Y == 2

i'm gonna assume you can plug p=3/4 in

Actually its 20 solutions.

ur right, I changed that :D

nvm my friend told me 

correct though you should say it's farther away from the center, or from (0,0)

\(\text{let }P[\text{heads}] = p\\ E[V] = 2p -(1-p) = 3p-1\)

I suspect there were numbers for p and (1-p) but they didn't come through.

Sorry, no time to explain.

(6y^5)/(x^3)

We  have this  arithmetic  series

10  +  12  + 14 + ......+  ???  =  220

The sum of an   arithmetic series is given  by :

S  =  (n / 2)  [ 2a  +  (n - 1) * d ]    

Where a  is the first term, n is the number of terms  [ and the number of rows in this case ] and  d  is the difference between terms

So we have

(n / 2)  [ 2(10)  + ( n - 1) * 2 ]   =  220        multiply both sides by 2

n [ 20 + 2n - 2 ]   =  440

n [ 18 + 2n ]  = 440

2n^2 + 18n - 440  =  0      divide through by 2

n^2 + 9n - 220  =  0       factor

(n + 20) ( n - 11)  =  0

Set eacn  factor to 0  and solve for n  and we get that

n = -20    [ reject ]    or

n  = 11    [ accept ]

So....11 rows

There should be 11 solutions

sqrt [ 4  +  sqrt (63)  / 2 ]  + sqrt [ 4 - sqrt (63 /2) ]

Square this

4 + sqrt (63/2)  +  2 sqrt [ 4  +  sqrt (63)  / 2 ]  * sqrt [ 4 - sqrt (63 /2) ] + 4 - sqrt(63/2)

[ 4 + sqrt (63 / 2)   +    2 sqrt  [ 16 - 63 / 4 ]  + 4 - sqrt (63/2)  ]   =

 8  + 2 sqrt [( 64 - 63) / 4  ]  =

8 + 2 sqrt ( 1/4)  =

8 + 2 (1/2)  =  

9

Since both  quantities under the original radical are positive, take the  positive square root  of 9  =

3

We have the points   (0, 15)  and  ( 150, 0 )

Slope  =  [  0 - 15  ]  / [ 150 - 0 ]  =   -15 / 150   =  - 1 / 10

Equation using  ( 0 , 15)

y  = (-1/10) ( x - 0) + 15

y = (-1/10) x + 15     multiply  through by -10

-10y = x - 150     add 10y to both sides

0  = x + 10y - 150

1x + 10y - 150  =  0    ⇒  Standard Form

x^2  - 5x  - 8  = 0    add 8 to both sides

x^2  - 5x   = 8

Take  1/2 of 5  = 5/2....square it  = 25/4    .....add to both sides

x^2 - 5x  + 25/4  =  8 + 25/4       factor the left side, simpliify the right

(x - 5/2)^2  =  57/4      take both roots

x - 5/2  =  ± √ [ 57 / 4 ]

x - 5/2  = ± √57 / 2      add 5/2 to both sides

x  = 5/2  + √57/ 2          or   x  =  5/2   - √57 / 2

Thank you for being smart 

We  have the points  (0, 56)  and (1,70)

We want to work to the form  Ax  + By  + C  =  0

Slope  =  [ 70 - 56 ]  [ 1, 0 ] =  14 / 1  = 14

Using the point  (0, 56)  we have

y  = 14 (x - 0) + 56

y  = 14x + 56    subtract  y from both sides

14x - y + 56  = 0

14x - 1y +  56  =  0

Accumulatedl amount  =  P ( 1 + r/4)^(n *t)

P = 32000

r = 3%  = .03

n = number of compoundings per year  =  4

t = number of years  = 7

32000 ( 1 + 03 / 4)^28  = 

39,446.78

80 (1.25)^10  ≈

745  books at the end of the 10th month

Slope  =  [ 5 - 15 ] / [ 0 - 2 ]  =  -10  / -2   =  5

Using   (0,5)   we have

y  =  m ( x - x1)  + y1           where m is the slope  and (x1, y1)  = ( 0, 5)

y = 5 (x - 0)  + 5   

y = 5x + 5    in slope-intercept form