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By "grouping" and factoring:
8x^3-12x^2+2x-3=0
(8x^3-12x^2)+(2x-3)=0
4x^2(2x-3)+(2x-3)=0
factoring out the (2x-3):
(2x-3)(4x^2+1)=0
.
setting each term to zero:
2x-3=0
2x=3
x=3/2
and
4x^2+1 = 0
4x^2 = -1
x^2 = -1/4
x = sqrt(-1/4)
x = +-(1/2)i
.
solution:
x = (3/2, (1/2)i, -(1/2)i}

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The population of rabbits in an area is modeled by the growth equation P(t)=8e^.26t, where P is in the thousands and t is in years. How long will it take for population to reach 25,000?
.
Set P(t) to 25000 and solve for t:
P(t)=8e^(.26t)
25000 = 8e^(.26t)
25000/8 = e^(.26t)
3125 = e^(.26t)
ln(3125) = .26t
ln(3125)/.26 = t
8.047/.26 = t
30.95 = t
or approximately
31 years = t

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5.​A company that builds phones earns a profit on sales according to the quadratic relation
P = -10n2 + 1000n -2500
where P is the profit earned in dollars, and N is the number of crates of phones sold.
a. How many crates must be sold to make a maximum profit?
P = -10n^2 + 1000n -2500
Find derivative:
P' = -20n + 1000
Set P' to zero to find critical points:
0 = -20n + 1000
-1000 = -20n
1000 = 20n
100 = 2n
50 = n
.
b. What is the maximum profit that can be made?
plug n=50 and find P:
P = -10n^2 + 1000n -2500
P = -10(50)^2 + 1000(50) -2500
P = -10(2500) + 1000(50) -2500
P = -25000 + 50000 -2500
P = 25000 -2500
P = $22500
.
c. If you sell 11 crates how will you make a profit or a loss? How much?
plug n=11 and find P:
P = -10n^2 + 1000n -2500
P = -10(11)^2 + 1000(11) -2500
P = -10(121) + 11000 -2500
P = -1210 + 11000 -2500
P = 9790 -2500
P = $7290 (Profit)

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In a labatory, a culture increases from 30 to 195 organisms in 5 hours. What is the hourly growth rate in the growth formula y=a(1+r)^t
.
y=a(1+r)^t
The problem gives us:
y as 195
a as 30
t as 5
.
195=30(1+r)^5
195/30 = (1+r)^5
6.5 = (1+r)^5
6.5^(1/5) = 1+r
6.5^(.2) = 1+r
6.5^(.2) - 1 = r
1.454 - 1 = r
0.454 = r
45.4% = r

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During practice, a softball pitcher throws a ball whose height can be modeled by the equation h=-16t^2+24t+1, where h=height in feet and t=time in seconds. How long does it take for the ball to reach a height of 6 feet?
.
Set h to 6 and solve for t:
h=-16t^2+24t+1
6=-16t^2+24t+1
0 = -16t^2+24t-5
0 = 16t^2-24t+5
solve the above using the "quadratic formula" which yields:
t = {0.25, 1.25}
The ball reaches the height of 6 feet "on the way up" at
0.25 seconds
and again, "on the way down" at
1.25 seconds
.
Details of quadratic follows:

Solved by pluggable solver: SOLVE quadratic equation with variable

Quadratic equation (in our case ) has the following solutons: For these solutions to exist, the discriminant should not be a negative number. First, we need to compute the discriminant : . Discriminant d=256 is greater than zero. That means that there are two solutions: . Quadratic expression can be factored: Again, the answer is: 1.25, 0.25. Here's your graph:

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The form:
a+bi
where
'a' is the "real" part
and
'b' is the "imaginary" part
.
what is the conjugate of 8i in a+bi form
8i
in complex form is
0 + 8i
The conjugate is then
0+(-8)i
or
0-8i
or, simply
-8i
.
and the conjugate for 6.??
6
in complex form is
6 + 0i
conjugate is
6 - 0i
but, that's simply:
6

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24x to the 14th power square rooted.
.
Looks like:

we can break up the x's like this:

now, we can take "pairs" out from underneath the radical:

factoring the 24 to prime numbers:

now, take out one pair of 2's:

final answer:

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I'm trying to graph y=6 and x=-5, and I can't seem to find a website that can actually graph this. If you can help that would be great.
.
y=6
is a horizontal line that crosses the 6 on the y-axis
.
x=-5
is a vertical line that crosses the -5 on the x-axis

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which of the following equation represents a horizontal line through (5,-3)?
y = -3

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the interest on Rs3000 for one year is Rs210. what is the interest on RS 4500 for one year.
.
I = PRT
where
I is interest
P is principal
R is rate
T is time
.
From:"interest on Rs3000 for one year is Rs210"
210 = 3000R(1)
210 = 3000R
210/3000 = R
.07 = R
7% = R
.
Now, we can answer:
what is the interest on RS 4500 for one year.
I = PRT
I = 4500(.07)(1)
I = 4500(.07)
I = Rs315

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Principal $500 Rate 13% Time 1yr
.
A = P(1+r)^t
where
A is amount after time t (what we're looking for)
P is principal
r is rate
t is time (years)
.
A = 500(1+.13)^1
A = 500(1.13)^1
A = 500(1.13)
A = $565.00

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5. A gardener wants to divide a square piece of lawn in half diagonally. What is the length of the diagonal if the side of the square is 8 ft? Leave your answer in simplest radical form.
.
Let d = diagonal
then applying Pythagorean theorem we get

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f(x)=x+3 and g(x)= x^2-1
.
(f+g)(x) = x+3 + x^2-1
(f+g)(x) = x+2 + x^2
(f+g)(x) = x^2+x+2
so,
(f+g)(3) = 3^2+3+2
(f+g)(3) = 9+3+2
(f+g)(3) = 14
.
(f*g)(x) = (x+3)(x^2-1)
(f*g)(x) = x^3-x+3x^2-3
(f*g)(x) = x^3+3x^2-x-3
so,
(f*g)(-2) = (-2)^3+3(-2)^2-(-2)-3
(f*g)(-2) = -8+3(4)+2-3
(f*g)(-2) = -8+12+2-3
(f*g)(-2) = 3

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.

.

.

.

.

.

.

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11% of x = 33 find x
.
.11x = 33
x = 33/.11
x = 300

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without plotting the points how will you know if the coordinate lies on its quadrant ?
By the signs of the x,y coordinates:
(+ , + ) : quadrant 1
(- , + ) : quadrant 2
(- , - ) : quadrant 3
(+ , - ) : quadrant 4

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18% of 49
18/100 * 49
.18 * 49
8.82

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The width of a rectangular field is 8ft. shorter than its length. The length is represented by l.
a. Write an expression in terms of l that represents the perimeter of the rectangle.
Let L = length
then
L-8 = width
.
Perimeter = 2(L + L-8) = 2(2L-8) = 4L-16
.
b. Given that the length of the rectangular field is 32 ft., find the perimeter of the field.
Perimeter = 4L-16
Perimeter = 4(32)-16
Perimeter = 128-16
Perimeter = 112 feet
How many 4 ft. sections of fence are required to enclose the field?
112/4 = 108 sections
.
c. If the perimeter of the field is 80 ft., find the area enclosed by the fence. Support your answer.
Perimeter = 4L-16
80 = 4L-16
96 = 4L
24 feet = L (length)
.
width:
L-8 = 24-8 = 16 feet
.
Area = 24*16 = 384 square feet

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An oil tank has two inlet pipes and one outlet pipe. One inlet pipe can fill the tank in 12 hours, and the other inlet pipe can fill the tank in 20 hours. The outlet pipe can empty the tank in 10 hours. How long would it take to fill the tank with all three pipes open?
Let x = time (hours) it takes to fill
then
x(1/12 + 1/20 - 1/10) = 1
multiply both sides by 120: common denominator
x(10 + 6 - 12) = 120
x(16 - 12) = 120
x(4) = 120
x = 30 hours

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Factor 2x squared -1x-1
.

we can rewrite the middle term as:

grouping:

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if a tub take 10 min to fill and it takes 12min to drain how long will it take to fill up if the drain was left open ?
Let x = time (minutes) to fill
then
x(1/10 - 1/12) = 1
multiplying both sides by 120:
x(12 - 10) = 120
x(2) = 120
x = 60 minutes

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One maid can clean the house three times faster than another. Working together they can clean the entire house in three hours. How long would it take the faster maid cleaning alone?
Let x = time (hours) it takes faster maid
then
3x = time (hours) it takes slower maid
.
3(1/x + 1/(3x)) = 1
3(3 + 1) = 3x
3(4) = 3x
4 hours = x

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what is log base6 4 + log base 6 (x+2) = 1?
log base6 4 + log base 6 (x+2) = 1
log base6 4(x+2) = 1
4(x+2) = 6^1
4x+8 = 6
4x = -2
x = -1/2
.
what is log base2 (x+6)- log base2 (x) = 3?
log base2 (x+6)- log base2 (x) = 3
log base2 (x+6)/(x) = 3
(x+6)/(x) = 2^3
(x+6)/(x) = 8
(x+6) = 8x
6 = 7x
6/7 = x

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log(2x+8)=1+ log(x-5)
log(2x+8) - log(x-5) = 1
log(2x+8)/(x-5) = 1
(2x+8)/(x-5) = 10^1
(2x+8)/(x-5) = 10
(2x+8) = 10(x-5)
2x+8 = 10x-50
8 = 8x-50
58 = 8x
58/8 = x
7.25 = x

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General form:
y = a cos k(x-b)
where
amplitude is |a|
period is 2(pi)/k
phase shift is b
.
Your equation
y=3cos(2x-pi/2)
factor out the 2:
y=3cos 2(x-pi/4)
so, from inspection then
amplitude: 3
phase shift: (pi)/4
period:
(2(pi))/2
pi

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A rectangular garden has dimensions 3m by 4m. A path is built around the garden. The area of the garden and path is 6 times as great as the area of the garden. What is the width of the path?
.
Let x = width (m) of path
then
area of "path+garden" = (2x+3)(2x+4)
area of "garden" = 3*4 = 12
.
From:"The area of the garden and path is 6 times as great as the area of the garden." we get our equation:
(2x+3)(2x+4) = 6*12
4x^2+8x+6x+12 = 72
4x^2+14x+12 = 72
4x^2+14x-60 = 0
2x^2+7x-30 = 0
rewrite middle term:
2x^2+12x-5x-30 = 0
group:
(2x^2+12x)-(5x+30) = 0
2x(x+6)- 5(x+6) = 0
(x+6)(2x-5) = 0
x = {-6, 5/2}
we can throw out the negative solution (extraneous) leaving:
x = 5/2
or
x = 2.5 m

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An investment of $11,050 is divided into two accounts, one for stocks and one for mutual funds. The value of the stock account is $1,700 less than twice the value of the mutual fund account. Find the amount in the stock account.
.
Let x = amount ($) of stock account
then
11050-x = amount ($) in mutual funds
.
From:"The value of the stock account is $1,700 less than twice the value of the mutual fund account." we get our equation:
x = 2(11050-x) - 1700
x = 22100-2x - 1700
3x = 22100 - 1700
3x = 20400
x = $6800

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Volume of a sphere is:
(4/3)(pi)r^3
if you have HALF of a sphere:
(1/2)(4/3)(pi)r^3
(2/3)(pi)r^3
.
plug in our radius:
(2/3)(pi)(8)^3
(2/3)(pi)(512)
(1024/3)(pi) cubic feet

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Solve by the elimination method.
8x-y=69
x+9y=36
.
Multiply the top equation by 9 and combine both equations:
72x-9y=621
x+9y= 36
----------
73x =657
x = 9
.
Plug above into:
x+9y=36
9+9y=36
9y=27
y = 3
.
solution:
x=9 and y=3

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judy is Y years old. in 18 years she will be 3 times as old as she was 2 years ago. how old is she now?
.
Y+18 = 3(Y-2)
Y+18 = 3Y-6
18 = 2Y-6
24 = 2Y
12 years old = Y

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1. Which number makes this a true number sentence? Show your work.
397 + m > 412
397 + m - 397 > 412 - 397
m > 15
that is, m has to be "greater than" 15. Looking at the choice:
A. 14
B. 9
C. 16
D. 15
The ONLY one that is greater than 15 is is choice C
.
2. Which number does NOT make this a true number sentence? Show your work.
480 - z < 459
480 - z - 480 < 459 - 480
-z < -21
z > 21
Looking at available choices:
A. 25
B. 42
C. 21
D. 22
The ONLY choice is D