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log3(x-2)+log3(x+4)=3
log3[(x-2)(x+4)]=3
(x-2)(x+4)=3^3
x^2+4x-2x-8 = 9
x^2+2x-8 = 9
x^2+2x-17 = 0
.
Since we can't factor, we must use the quadratic equation.
Do so will yield:
x = {3.243, -5.243}
.
Details here:

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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=72 is greater than zero. That means that there are two solutions: . Quadratic expression can be factored: Again, the answer is: 3.24264068711928, -5.24264068711928. Here's your graph:

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5(5)^(-0.08x)=4
If so, start by dividing both sides by 5:
(5)^(-0.08x)=4/5
(5)^(-0.08x)=.8
Take the LOG5 of both sides:
-0.08x = LOG5(.8)
-0.08x = LOG(.8)/LOG(5)
-0.08x = -.0969/.6990
-0.08x = -.1386
x = 1.733

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Hey, this sounds familiar...
.
When you posted:
5(5)^-0.08x=6
I assume you mean:
5(5)^(-0.08x)=6
This means 5 is raised to the power of (-0.08x)
If this is incorrect, please write back
.
If so, start by dividing both sides by 5:
(5)^(-0.08x)=6/5
(5)^(-0.08x)=1.2
Take the LOG5 of both sides:
-0.08x = LOG5(1.2)
-0.08x = LOG(1.2)/LOG(5)
-0.08x = .0792/.6990
-0.08x = .1133
x = -1.416

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circumference of a circle:
2(pi)r
.
where pi is 3.14
.
30.2 = 2(3.14)r
15.1 = 3.14r
4.81 cm = r (radius)

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Area of any circle is:
(pi)r^2
.
They gave you the area as 547 and pi is 3.14. Plug and solve:
547 = (pi)r^2
547 = (3.14)r^2
174.2 = r^2
13.2 m = r (radius)

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log(x^2-x-5)= 0
x^2-x-5= 10^0
x^2-x-5= 1
x^2-x-6= 0
(x-3)(x+2)=0
.
x = {-2, 3}

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When you posted:
5(5)^-0.08x=6
I assume you mean:
5(5)^(-0.08x)=6
.
If so, start by dividing both sides by 5:
(5)^(-0.08x)=6/5
(5)^(-0.08x)=1.2
Take the LOG5 of both sides:
-0.08x = LOG5(1.2)
-0.08x = LOG(1.2)/LOG(5)
-0.08x = .0792/.6990
-0.08x = .1133
x = -1.416

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4log(2x+6)=8
log(2x+6)=2
2x+6=10^2
2x+6=100
2x=94
x = 47

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There is a class. The ratio is 5 boys to 4 girls. There are 65 boys. How many girls are there?
.
Let g = number of girls in the class
.
4:5 = g:65
or
4/5 = g/65
g = (4*65)/5
g = 260/5
g = 52 (girls in the class)

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.
Common denominator is:(k-3)(k+3)
.

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2/3 divided by x = 1
(2/3)/x = 1
multiplying both sides by x:
2/3 = x

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Charlie owns a coffee shop. He wants to create special mixes using 2 coffees,one priced at $6.40 per pound and the other priced at $7.28 per pound. How many pounds of the $7.28 coffee should he mix with 9 pounds of the $6.40 coffee to sell the mixture for $6.95 per pound?
.
Let x = pounds of $7.28 per pound coffee added
.
7.28x + 6.40(9) = 6.95(x+9)
7.28x + 57.6 = 6.95x + 62.55
0.33x + 57.6 = 62.55
0.33x = 4.95
x = 15 pounds (of $7.28 coffee)

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3+ Log(2x+5)= 2
Log(2x+5)= -1
2x+5= 10^-1
2x+5= 1/10
20x+50= 1
20x= -49
x = -49/20
x = -2.45

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A high school sold gift wrap. The gift wrap in solid colors sold for $4.00 per roll and the print gift wrap sold for $6.00 per roll. The total amount of rolls sold was 480, and the total amount of money collected was $2,340.00. How many rolls of each kind of gift wrap was sold?
.
Let x = number of $4 rolls sold
then
480-x = number of $6 rolls sold
.
4x + 6(480-x) = 2340
4x + 2880 - 6x = 2340
2880 - 2x = 2340
2880 = 2x + 2340
540 = 2x
270 = x ($4 rolls sold)
.
480-x = 480-270 = 210 ($6 rolls sold)

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Simply isolate W to one side of the equation:
XW + Y = B/4
XW = (B/4) - Y
W = (B/4X) - (Y/X)
.
The only restriction is:
Except when X=0
.
Notice that if X was equal 0, the numerator would be zero and a zero in the denominator would be "undefined".

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If Leah is 6 years older than her sister, Sue, and John is 5 years older than Leah, and the total of their ages is 41. Then how old is Sue?
.
Let x = Sue's age
then
x+6 = Leah's age
x+6+5 = John's age
.
x + x+6 + x+6+5 = 41
3x + 17 = 41
3x = 24
x = 8 years (Sue's age)

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Multiplying both sides by (x-6):

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The Krunch Cereal Co. changed the size of its box. The old box was 3 by 8 by 10. The new box is 4 by 8 by 12. Find the percent of change in the volume of the box.
.
Old box volume:
3*8*10 = 240
.
New box volume:
4*8*12 = 384
.
Volume change:
384/240 * 100
= 160%

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Refer to this site for a good explanation:
http://www.purplemath.com/modules/drofsign.htm
.
f(x)=-2x^5+3x^4-6x^2+1
There are three sign changes.
So, 3 or 1 positive roots
.
f(-x)=-2(-x)^5+3(-x)^4-6(-x)^2+1
f(-x)=2(x)^5+3(x)^4-6(x)^2+1
There are two sign changes.
So, 2 or 0 negative roots
.

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As you say:
(x-5)(x-1)
x^2 - x - 5x + 5
x^2 - 6x + 5
.
So,
F(x) = x^2 - 6x + 5
.
F(1) = 1^2 - 6(1) + 5
F(1) = 1 - 6 + 5
F(1) = 0
.

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.
Slope between two points, can be found by:
m = (y2-y1)/(x2-x1)
.
We can pick any two points but we'll pick 1 and 2 to determine the slope:
m = (y2-y1)/(x2-x1)
m = (20-24)/(2-1)
m = (-4)/(1)
m = -4
.
Now, we can plug any point, say (5,8), and the slope (-4) and plug it into the "point-slope" form:
y-y1 = m(x-x1)
y-8 = -4(x-5)
y-8 = -4x + 20
y = -4x + 28 (this is what they're looking for)

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You don't say what the problem is asking for, but I suspect they want to know the amount (x) when supply equals demand.
.
If so, set the two given equations equal to each other and solve for 'x':
5000 + 200x = 9500 - 100x
5000 + 300x = 9500
300x = 4500
x = $15

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Use long division to find the quotient Q(x) and the remainder R(x) when P(x) is divided by d(x).
P(x)=x^4+2x^3-2x+4
d(x)=x-2
.
NOTE: it will be easier to see the solution (because of the spacing) below if you view the "source"!
.
x^3+5x^2+2x+2
_______________
x-2 |x^4+2x^3+0x^2-2x+4
x^4-3x^3
---------
5x^3+0x^2-2x+4
5x^3-2x^2
--------------
2x^2-2x+4
2x^2-4x
---------
2x+4
2x-4
----
8
.
Therefore:
Q(x) = x^3+5x^2+2x+2
R(x) = 8/(x^4+2x^3+0x^2-2x+4)

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fran swims at a speed of 2.2 km/h in still water. the lazy river flows at a speed of 0.4 km/h. how long will it take fran to swim 0.9 km upstream?
.
We will need to apply the "distance" formula:
d = rt
where
d is distance
r is rate or speed
t is time
.
Reordering the equation, we solve for 't':
t = d/r
.
Fran's rate (r) is:
2.2 - 0.4 = 1.8 km/h
.
Therefore:
.9/1.8 = 0.5 hours

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Rental Company A charges $12.00 plus 80 cents per hour to rent a paint sprayer. A competitor, Rental company B rents the same sprayer for $3.95 plus $1.50 per hour.
Write an algebraic equation for the cost in terms of the number of hours rented for each business.
Determine algebraically the number of hours for which the cost of renting a paint sprayer will be the same for company A or B
.
Let h = number of hours rented
.
Company A:
C(h) = 12 + .80h
.
Company B:
C(h) = 3.95 + 1.50h
.
To find out when the cost will be the same, set it equal to each other and solve for 'h':
12 + .80h = 3.95 + 1.50h
12 = 3.95 + .70h
8.05 = .70h
11.5 hours = h

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You can find your answer at:
http://www.csgnetwork.com/stopdistcalc.html

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1st one: A lot is in the shape of a right triangle. The shorter leg measures 180 m. The hypotenuse is 60 m longer than the length of the longer leg. How long is the longer leg??
.
Let x = length of longer leg
then
x+60 = hypotenuse
.
Applying pythagorean's theorem:
180^2 + x^2 = (x+60)^2
180^2 + x^2 = (x+60)^2
32400 + x^2 = x^2 + 120x + 3600
32400 = 120x + 3600
28800 = 120x
240 meters = x (longer leg)
.
2nd one: A rectangular piece of cardboard measuring 28 inches by 29 inches is to be made into a box with an open top by cutting equal size squares from each corner and folding up the sides. Let x represent the length of a side of each such square. For what value of x will the volume be a maximum? If necessary, round to 2 decimal places.
.
Volume = x(28-2x)(29-2x)
Volume = x(812-56x-58x+4x^2)
Volume = x(812-114x+4x^2)
Volume = 812x-114x^2+4x^3
Volume = 4x^3-114x^2+812x
.
Taking the derivative:
4x^3-114x^2+812x
12x^2-228x+812 = 0
6x^2-114x+406 = 0
Solving the above with the quadratic equation we get:
x = {14.25, 4.75}
.
Solution: 4.75 inches
.
Details of quadratic equation:

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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=3252 is greater than zero. That means that there are two solutions: . Quadratic expression can be factored: Again, the answer is: 14.2521924764611, 4.74780752353892. Here's your graph:

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3(x-2)^2=48
Dividing both sides by 3:
(x-2)^2=16
Take the square root of both sides:
x-2 = (+-)4
Add 2 to both sides
x = 2 (+-)4
or
x = {-2, 6}

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2x+5y=3
5x-2y=1
.
Convert both to the form of:
y = mx + b
to determine the slope of each line.
.
2x+5y=3
5y=-2x+3
y=(-2/5)x + (3/5)
slope is -2/5
.
5x-2y=1
2y = 5x + 1
y = (5/2)x + (1/2)
slope is 5/2
.
Since multiplying both slopes together:
(-2/5)(5/2) = -1
they are negative reciprocal
therefore the two lines are perpendicular

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First, find the distance between the center and the bisected point is:
Do this using the "distance formula" between two points:
d = sqrt[(x2-x1)^2 + (y2-y1)^2]
(-3,-4) and (-11/12, -13/12)
d = sqrt[(-11/12+3)^2 + (-13/12+4)^2]
d = sqrt[(-11/12+36/12)^2 + (-13/12+48/12)^2]
d = sqrt[(25/12)^2 + (35/12)^2]
d = sqrt[4.34 + 8.51]
d = sqrt[12.85]
d = 3.584
.
Drawing a diagram of the problem will help you see that the "distance between the center and bisector", "half the chord" and the "radius" forms a right triangle -- allowing you to apply the pythagorean theorem:
Let x = half the length of the chord
then
x^2 + 3.584^2 = 5^2
x^2 + 12.847 = 25
x^2 = 12.153
x = 3.486
.
Length of chord is
2x = 2(3.486) = 6.972

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Simply, divide 5 by 8 either on a calculator or by long-hand.
.
5/8 = 0.625
.
Write back if this doesn't make sense.