CPhill

5+(0.3)^2 =

5 + (3/10)^2  =

5 + 9/100  =

5.09    in  decimal form

y - 1   = x    →  y = x + 1    (1)

x^2 + y^2 = 17   (2)

Subbing (1) into (2), we have

x^2 + ( x + 1)^2  = 17       simplify

x^2 + x^2 + 2x + 1  = 17

2x^2 + 2x - 16  = 0       divide through by 2

x^2 + x - 8  = 0

This will not factor.......using the quadratic formula, the two solutions for x are

[ -1 + sqrt(33)] / 2       and   [ -1 - sqrt(33)] / 2

And y = x + 1

So.....when x = [ -1 + sqrt(33)] / 2 , y = x + 1 =  [1 + sqrt(33)]/2

And, when x = [ -1 - sqrt(33)]/ 2 , y = x + 1 =   [1 - sqrt(33)] / 2

And the sum of all these solutions is

[ -1 + sqrt(33)] /2   +   [ -1 - sqrt(33)] / 2   +  [1 + sqrt(33)]/2 + [1 - sqrt(33)] / 2  = (suprisingly, 0  !!! )

You can see why this is so from the following graph: https://www.desmos.com/calculator/8axms04lcy

The solution coorrdinates just "cancel" each other out when we sum them !!!!!! 

The sum of the first N positive integers is given by :

(N) (N + 1)  / 2    ........so we have....

(366) (367) / 2    =  67,161

The common ratio is given by the ratio of the following two terms :

a_{n+ 1} / a_n  

So.....we have

  -3 / 9   = -1/3 = the common ratio    ......verify for yourself that this ratio is consistent with the rest of the terms  

120  + 30x >= 310     subtract 120 from both sides

30x >= 190     divide both sides by 30

x >= 6 + 1/3 yards

So......6 yards won't be quite enough.......he needs to mow 7

No prob......

150 / 8  =  75 / 4   =  18 + 3/4

Sure !!!......consider....

(1/2)^2  = 1/4       and  1/4 < 1/2

In fact, if the base is  > 0 but < 1, this will always happen.....!!!!

3x - y = 6       (1)

xy =  24 →  y = 24/x    (2)

So.....  substituting (2) into (1), we have

3x - 24/x = 6    multiply through by x

3x^2 - 24 = 6x      re-arrange

3x^2 - 6x -24  = 0       divide through by 3

x^2 - 2x - 8  = 0     factor

( x - 4) ( x + 2)  = 0       and setting each factor to 0, we have

x = 4   and x = -2

So.... y = 24/4  = 6      or.... y = 24/-2  = -12

And we have   (4, 6)  and ( -2, -12)

And the sum of these  is just     10 + -14   = -4

Just perform  the operation in the brackets and add the % sign, Katie...[ (43 / 860)  x  100 ] %