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i upvoted your posts ;)

now pls tell me the answer so i can work backwards

o.o

because you are saying that us two are wrong when we are right.

unless you mean when c=c^2-2

what is the answer?

if you have anything else to add, pls add it!

this means that a=-c^2-c+2 (vieta's) and b=c^3-2c (also vieta's)

now you can see that all you need to do is plug in any value for c to get any two values for a and b. this means that there are infinitely many solutions to the equation.

i think you might have messed up ni the final steps.

x               x^2 - 6x + 8           x - 2

0                     8                      -2

1                     3                      -1

2                     0                       0

So....x = 2 is the solution

first find the ax intercept of the line. this is just 5*3/2 or 7.5, and since the slope is 2/3, then the height is 2/3 of the width of the triangle.

the width of the triangle is |k+7.5|, so the height is 2/3|k+7.5|. multiplying these, we can take away the absolute value because a square is always non negative, so you get ((k+7.5)^2)/3 because you have to divide by 2 because of a triangle. this is less than 20, so (k+7.5)^2 is less than 60, so k can be 0,-1,-2,-3,-4,-5,-6,-7,-8,-9,-10,-11,-12,-13,-14, and -15. adding these up you get -120.

HOPE THIS HELPED!

also cphill, i think you might have forgotten that a triangle is width*height/2.

edit: nevermind, but you are wrong, i am right. i dont know where you made a mistake though, i graphed my answers on desmos, they work.

confused??!

The line intersects the x axis at x = -7.5

The base length of the triangle will be   k - (-7.5) =  k + 7.5

The height of the triangle will be    (2/3)k +  5

So...we are trying to solve this

(1/2) base * height = 20

(1/2) ( k + 7.5) [ (2/3)k + 5]  = 20

(k + 7.5) (2/3k + 5)  =  40

(2/3)k^2 + 5k + 5k + 37.5 = 40

(2/3)k^2 + 10k - 2.5 = 0             multiply through by 3

2k^2 + 30k - 7.5   =  0           multiply through by 10

20k^2 + 300k - 75  = 0

4k^2 - 60k - 15 = 0

The soutions to this are   k ≈ -15.246      and k ≈ .245   

So....the integer values for k that make this triangle have an area < 20    are  the integers  -15 to 0 inclusive

And the sum of these =  - ( 15) (16) / 2  =  -120 

The 15th term ends with 2

\(\text{You want the expected value of the amount of money that will be made}\\ E[M] = (0.35)(-\$15000) + (0.4)(\$45000) = \$12750\\ \text{The expected value shows a profit and thus the investment should be made}\)

\(x^2+ax+b = (x-c)(x-(c^2-2) = \\ (x-c)(x-c^2+2) = \\ x^2 +(2-c-c^2)x + (c^3-2c) \\\)

\(\text{As we don't have any limitations on }c\\ \text{There are infinitely many equations where both }c \text{ and }c^2-2\\ \text{are both roots, given by }x^2+ax+b \\ a = 2-c-c^2\\b = c^3-2c\)

What does the 15th term end in??!!

\(5xy+3xy^2 = 7\\ x(5y+3y^2)=7\\ \text{we know that }5y+3y^2 \neq 0 \text{ as it's a factor of a non-zero number (7), so}\\ x = \dfrac{7}{5y+3y^2}\)

This is simple pattern recognizing

the first few terms are 1, 2, 2, 4, 8, 32, 256, 8192, 2097152...

Notice how the last digits are forming a pattern, (2, 2, 4, 8, 2, 6, 2, 2, 4, 8, 2, 6, 2...

The pattern is 2 -> 2 ->  4 -> 8 ->  2 -> 6

So if the pattern repeats every 6 terms. And assuming you are trying to find the 14th term. You divide 14 by 6 and find the remainder, which is 2.

So look at the 2nd term in the pattern. which is 2.

But remember the first term is 1, so you have to count back a term

So the answer is 2

This  is a "cubic equation" not a "qudratic equation".
You could use the "cubic formula" which is analogous to "quadratic formula", which will give you the same results:
x = 3.41421 3562     =2 + sqrt(2)
x = 0.58578 64376   =2 -  sqrt(2)
x = 1.66666 6667     =5/3

I assume he/she means the 14th term. If so, then the 14th is:

1
2
2
4
8
32
256
8192
2097152
1 7179869184
3602879 7018963968
6189700 1964269013 7449562112
2230 0745198530 6231415357 1827264836 1505980416
1 3803492693 5811275748 6951172455 4050904902 2179443407 7311032504 8447598592

last digit of which term?

Wait but if you use quadratic formula it simplifies out to 2+-2*sqrt(2) all over 2. so the two 2's would cancel out and you would be left with:

x = 5/3 or x = sqrt 2 or x = -sqrt 2

Solve for x:
3 x^3 - 17 x^2 + 26 x - 10 = 0

The left hand side factors into a product with two terms:
(3 x - 5) (x^2 - 4 x + 2) = 0

Split into two equations:
3 x - 5 = 0 or x^2 - 4 x + 2 = 0

Add 5 to both sides:
3 x = 5 or x^2 - 4 x + 2 = 0

Divide both sides by 3:
x = 5/3 or x^2 - 4 x + 2 = 0

Subtract 2 from both sides:
x = 5/3 or x^2 - 4 x = -2

Add 4 to both sides:
x = 5/3 or x^2 - 4 x + 4 = 2

Write the left hand side as a square:
x = 5/3 or (x - 2)^2 = 2

Take the square root of both sides:
x = 5/3 or x - 2 = sqrt(2) or x - 2 = -sqrt(2)

Add 2 to both sides:
x = 5/3 or x = 2 + sqrt(2) or x - 2 = -sqrt(2)

Add 2 to both sides:

x = 5/3     or      x = 2 + sqrt(2)      or      x = 2 - sqrt(2)