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Find in standard form, the equation of an ellipse whose center is at (2,-1) whose major axis of length 10 is along the y-axis, and whose minor axis of length 8 is along the x-axis.
**
Given equation is that of an ellipse with a vertical major axis.
Its standard form: , a>b, (h,k)=(x,y) coordinates of center.
..
Given center: ((2,-1)
Given length of vertical major axis=10=2a
a=5
a^2=25
given length of minor axis=8=2b
b=4
b^2=16
Equation:

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What are the asymptotes of the graph of y = 2 divided by x+18?
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To find the asymptote of a rational expression, set the denominator =0, then solve for x.
At this value of x, the function becomes undefined
for given problem, 2/(x+18).
x+18=0
x=-18
equation of asymptote: x=-18

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As part of flight training, a student pilot was required to fly to an airport and then return. The average speed on the way to the airport was 125 mph, and the average speed returning was 218.75. Find the distance between the two airports if the total flying time was 8.25 hours.
**
let x=distance between airports
travel time=distance/speed
..

LCD:125*218.75
218.75x+125x=8.25*125*218.75
343.75x=8.25*125*218.75
x=656.25
distance between airports=656.25 miles

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Having a problem figuring out this algebra. Any help will be appreciated thanks
(4+8i)(4-8i)
**
You should recognize this the factored form of the difference of two squares which go like this:
(a+b)(a-b)=a^2-b^2
..
(4+8i)(4-8i)
=4^2-(8i)^2
=16-64i^2
i^2=-1
=16+64
=80

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a wire is attached to a cell phone tower at a point 210 feet above the ground. company policy requires the slope of the wire to be (7/3). Hw far from the base of the tower does the wire attach to the ground?
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let x=distance from base of tower to where wire attaches to the ground
Slope is always expressed as a ratio, ∆y/∆x=change in y/change in x
We then apply this ratio to the given problem.
210/x=7/3
7x=3*210
x=630/7=90
distance from base of tower to where wire attaches to the ground=90 ft

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i don't understand how to do a polynomial equation, particularly ones with exponents higher then 2 like x^3+9x^2+23x+15=0
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f(x)=x^3+9x^2+23x+15=0
You can try to factor the expression by grouping, but if this is difficult like given equation, then you must use the rational roots theorem as follows:
Using synthetic division:
...0..|.....1......9.......23.......15
.-1..|.....1.... .8.......15........0 (-1 is a root)
f(x)=(x+1)(x^2+8x+15)=(x+1)(x+5)(x+3)
zeros are: -5,-3, -1
note: you can always check your answer on a graphics calculator like a TI-83+

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Find a focal point of the ellipse:
((x+3)^2)/16)+(y-1)^2)/4)=1
This is an equation of an ellipse with horizontal major axis:
Its standard form: (x-h)^2/a^2+(y-k)^2/b^2=1, (a>b), (h,k)=(x,y) coordinates of center
For given ellipse:
center(-3,1)
a^2=16
a=√16=4
b^2=4
b=√4=2
c^2=a^2-b^2=16-12=4
c=√4=2
Foci: (Both focal points)=(-3±c,1)=(-3±2,1)=(-5,1) and (-1,1)
answer a is the correct answer

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A ladder is leaning agains a building. The distance from the bottom of the ladder to the building is 2 ft shorter than the length of the ladder. How high up the side of the building is the top of the ladder if the distance is 1 ft less than the length of the ladder?
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let x= length of ladder
x-2=distance from bottom of ladder to bldg
x-1=How high up the side of the building is the top of the ladder
This makes for a right triangle with legs, (x-1), (x-2) and hypotenuse x.
..
By Pythagorean Theorem:
x^2=(x-1)^2+(x-2)^2
x^2=(x^2-2x+1)+(x^2-4x+4)
x^2=2x^2-6x+5
x^2-6x+5=0
(x-5)(x-1)=0
x=1 (reject, not reasonable)
or
x=5
x-1=4
How high up the side of the building is the top of the ladder: 4 ft

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a chemist added 2L of water to 8L of a 48% nitric acid solution. What is the acid concentration of the new solution?
**
let x=acid concentration of new solution
48%*8=x(10+2)=12x
12x=3.84
x=3.84/12=.32
acid concentration of new solution=32%

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How do you simplify
√16x^5 + y^4 + z^2
=4x^2√x+y^4+z^2
that is all you can do

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Can some one please help me to solve these radical expression and help me to decide if they have a solution. If the radical has no solution can you please explain why? Thanks in advance for your help.
1. Solve. Sqrt x=-3 If the equation does not have a solution, explain why.
√x=-3
square both sides
x=9
..
2. Solve. Sqrt x+5= Sqrt 2x+8. If the equation does not have a solution, explain why.
√(x+5)=√(2x+8)
x+5=2x+8
2x-x=5-8
x=-3
..
3. Solve. x-3= Sqrt 17-4x. If the equation does not have a solution, explain why.
(x-3)=√(17-4x)
square both sides
x^2-6x+9=17-4x
x^2-2x-8=0
factor
(x-4)(x+2)=0
x=4
or
x=-2

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find all zeros of the following polynomial write the polynomial in factored form
f(x) = 7x^3-x^2+7x-1
factor by grouping
f(x) =x^2(7x-1)+(7x-1)
f(x) = (7x-1)(x^2+1)=0
7x-1=0
x=1/7
or
x^2+1=0
x^2=-1
x=±√-1=±
zeros: 1/7 and two non-real zeros, ±i

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Solve the polynomial inequality and graph the solution set on a real number line. Express the solution set in interval notation.
x^3-4x^2-16x+64<0
factor by grouping
x^2(x-4)-16(x-4)<0
(x-4)(x^2-16)<0
(x-4)(x+4)(x-4)<0
number line
<....-....-4.....+.....4....+.....>
solution: (-∞,-4)

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two angles are supplementary. one angle is 40degrees more than four times the other. find the measures of the angle. its complement and its supplement
**
let x=one of two supplementary angles
180-x=other supplementary angle
..
180-x=4x+40
5x=140
x=28
180-x=152
90-x=62
..
measure of the angle=28º
Its complement=62º
Its supplement=152º

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factorise by using factor theorum :- y^3-7y+6 (please show its steps also)
**
f(y)=y^3-7y+6
Use rational roots theorem to solve: (with synthetic division)
...0..|...-1......0......-7.......6............
...1..|.....1......1......-6......0 (1 is a root)
f(y)=(y-1)(y^2+y-6)
f(y)=(y-1)(y+3)(y-2)

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i have a problem of this question factorize it a3-13a+12.
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f(a)= a3-13a+12
Use rational roots theorem to solve: (with synthetic division)
...0..|.....1......0......-13.......12............
...1..|.....1......1......-12........0 (1 is a root)
f(a)=(a-1)(a^2+a-12)
f(a)=(a-1)(a+4)(a-3)

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how much each of 20% and 80% alcohol should be added to get 5 gallons of a 50% alcohol mixture
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let x=amt of 20% alcohol to add
5-x=amt of 80% alcohol to add
..
20%x+80%(5-x)=50%*5
.2x+4-.8x=2.5
.6x=1.5
x=2.5
5-x=2.5
..
amt of 20% alcohol to add=2.5 gal
amt of 80% alcohol to add=2.5 gal

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What is the measures of two complementary angles such that the larger angle is six less than twice the smaller angle?
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let x=larger of two complementary angles
90-x=smaller of two complementary angles
..
x=2(90-x)-6
x=180-2x-6
3x=174
x=58
90-x=32
larger of two complementary angles=58º
smaller of two complementary angles=32º

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A woman invests money in two funds that yield an annual total income of $86. One fund pays an annual return of 7%, and the other pays a return of 8%. How much is invested in the fund yielding 7% if the total annual amount invested at the start of the year is $1200.00
**
let x=amt invested in 7% fund
1200-x=amt invested in 8% fund
..
.07x+.08(1200-x)=86
.07x+96-.08x=86
.01x=10
x=1000
1200-x=200
amt invested in 7% fund=$1000

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steps to solve x^4 + 2x^2-15=0
factor
(x^2+5)(x^2-3)=0
x^2+5=0
x^2=-5
x=±√-5
x=±(√5) i
or
x^2-3=0
x^2=3
x=±√3
two real roots: ±√3
two imaginary or non-real roots: ±(√5) i

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Write your solution using set builder notation. Please show all of your work.
solve
-11x-24<9-2(5x-9)
-11x-24<9-10x+18
-x<24+27
-x<51
x>-51
{x|x>-51}

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When a stone is dropped into a deep well, the number of seconds t until the sound of a splash is heard is given by the formula
T=√x/4+x/1100
where x is the depth of the well in feet. If it takes 9 seconds to hear a splash, how deep (to the
nearest foot) is the well?
T=√x/4+x/1100
9=√x/4+x/1100
LCD:4*1100
1100√x+4x=9*4*1100=39600
1100√x+4x-39600=0
275√x+x-9900=0
let u=√x
u^2=x
..
u^2+275u-9900=0
Solve by following quadratic formula:

a=1, b=275, c=9900
ans:
u=32.2=√x
square both sides
x≈1037 ft

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|-11|+5²=50y-|+31|-49y
The absolute value of any number is always positive.
11+5²=50y-31-49y
11+25=50y-31-49y
y=11+25+31=67

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a car leaves Rockford, Il at teh rate of 65mph bound for Madison,WI. At the same time, a truck leaves Madison bound for Rockford at the rate of 55 mph. If the cities are 72 miles apart how long will it take for the car and truck to meet?
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let x=distance car traveled
72-x=distance truck traveled
travel time=distance/speed
65/x=55/(72-x)
55x=65*72-65x
120x=65*72
x=65*72/120=39 mi
72-x=33 mi
travel time=39/65=0.6 hr or 36 min
time car and truck met: 36 min

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The perimeter of a rectangle is 28 cm. if the length of a diagonal is 10 cm, find the length and width of the rectangle.
**
let x=length of rectangle
let y=width of rectangle
2x+2y=perimeter=28
x+y=14
y=14-x
by pythagorean theorem:
x^2+y^2=10^2=100
x^2+(14-x)^2=100
x^2+196-28x+x^2=100
2x^2-28x+196=100
2x^2-28x+96=0
x^2-14x+48=0
(x-8)(x-6)=0
x=8
or
x=6
length of rectangle: 8 cm
width of rectangle: 6 cm

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Consider the line 8x-7y=-1, find the equation of the line that is parallel to this line and passes through the point (1,3). Find the equation of the line that is perpendicular to this line and passes through the point (1,3)
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Standard form of equation for a straight line: y=mx+b, m=slope, b=y-intercept
Original line:
8x-7y=-1
7y=8x+1
y=8x/7+1/7
slope=8/7
..
Perpendicular line:
slope:-7/8 (negative reciprocal of slope of original line.
Equation: y=-7x/8+b
solving for b using given point (1,3)
3=-7*1/8+b
b=3+7/8=31/8
equation: y=-7x/8+31/8
..
Parallel line:
slope: 8/7 (same as slope of original line)
Equation: y=8x/7+b
solving for b using given point (1,3)
3=8*1/7+b
b=3-8/7=13/7
equation: y=8x/7+13/7
see graph below as a visual check:
red line:original
blue line: parallel
green line: perpendicular

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The frame on a picture is 28 cm by 32 cm outside and is of uniform width. What is the width of the frame if 192 cm^2 of the picture shows
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let w=width of the frame
Area of frame+picture=28*32=896
Area of frame=896-192=704=length*w=2*(28+32)*w
w=704/2*(28+32)=704/120≈5.87
width of the frame≈5.87 cm

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Hello! I need help solving 2 equations involving Logarithms.
How do I correctly graph the equation bellow on the same coordinate plane.
..
Question #1: F(x)= 2^-x and G(x)= 4^x
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For F(x)=2^-x=1/2^x
Domain: (-∞,∞)
Range: (0,∞)
Horizontal asymptote: x-axis or y=0
..
For F(x)=4^x
Domain: (-∞,∞)
Range: (0,∞)
Horizontal asymptote: x-axis or y=0
See graph below: Green curve is 4^x and red curve is 2^-x or 1/2^x
Note that both curves have a y-intercept=1 (2^0=1)
Also note both functions are always >0
..
Question #2:
How do I solve/or rewrite this equation as a single quantity?
log^2 (x) + log^2 (x+2) = log^2 (x +6)
log^2[ (x) (x+2)] = log^2 (x +6)
x(x+2)=x+6
x^2+2x=x+6
x^2+x-6=0
(x+3)(x-2)=0
x=-3 (reject, x>0)
or
x=2

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Describe the following characteristics of the logarithmic function y = logax.
1) Domain: (0,∞)
2) Range: (-∞,∞)
3) x-intercept: 1
4) Horizontal asymptote: none
5) Vertical asymptote: y-axis or x=0
6) General shape: starting from (0,-∞) curve becomes less negative until it crosses the x=axis at (1,0), at which point the curve becomes increasingly positive to ∞.
Number line:
<...-....1...+.....>

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Write the expression as a single logarithm with a coefficient of 1. Assume all variable expressions represent positive real numbers.
ln(6x + 7) − ln y − 8 ln z + (1/2)ln w
ln(6x + 7) −(ln y + 8 ln z) + (1/2)ln w
ln[(6x+7)(w^1/2)/ y*z^8]

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Write the expression as a single logarithm with a coefficient of 1. Assume all variable expressions represent positive real numbers.
logb(3x) + logb(y + 4) + logb(y + 2) − logb(y^2 + 6y + 8)
logb(y^2 + 6y + 8)=log[(y+4)(y+2)]
logb[(3x(y+4)(y+2))/(y+4)(y+2)]
(y+4)(y+2) cancels out, leaving:
logb(3x)