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solve the equation using the quadratic formula method:3a^2x^2-abx-2b^2=0
use following quadratic formula to solve:

a=3a^2, b=-ab, c=-2b^2
I will let you take it from here

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(2)/(3t)^(1/2)
(2)/√(3t)
multiply top and bottom by √(3t)
(2)/√(3t)*√(3t)/√(3t)
2√(3t)/3t

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Find the value of k so that the three points lie on the same line. Write the equation in point slope form. The points given are (1,-2) (-2,4) (4,k)
**
y-y1=m(x-x1)
using given point (1,-2)
y-(-2)=m(x-1)
y+2=m(x-1)
using given points (1,-2) and (-2,4) to find slope,m
m=∆y/∆x=(4-(-2))/(-2-1)=(4+2)/-3=6/-3=-2
y+2=-2(x-1)
y+2=-2x+2
y=-2x
solving for k
k=-2*4=-8
k=-8
see graph below:

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you can travel 40 miles on motorcycle in the same time that it takes to travel 15 miles on bicycle. if your motorcycle's rate is 20 miles per hour faster than your bicycle's, fine the average rate for each
let x=rate of speed of bicycle
x+20=rate of speed of motorcycle
travel time= distance/rate of speed (same for both)
40/(x+20)=15/x
40x=15(x+20
40x=15x+300
25x=300
x=12
x+20=32
rate of speed of bicycle=12 mph
rate of speed of motorcycle=32 mph

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Find an equation of the line passing (-7,2) and (4,5)
Equation of a line: y=mx+b, m=slope, b=y-intercept
For given problem:
m=∆y/∆x=(5-2)/(4-(-7))=3/11
equation: y=3x/11+b
solve for b using one of given coordinates(4,5)
5=3*4/11+b
b=5-12/11
b=55/11-12/11=43/11
equation: y=3x/11+43/11
see graph below:

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Find all of the exact solutions to the following equation. (Use the parameter k as necessary to represent any integer.)
cos(3x) =1/2
3x=π/3+2πk, 5π/3+2πk, k=any integer
x=π/9+2πk, 5π/9+2πk, k=any integer

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Solve the following exponential equation. Exact answers only.
49^x-12*7^x=-36
7^2x-12*7^x=-36
let u=7^x
u^2=7^2x
u^2-12u+36=0
(u-6)(u-6)=0
u=6=7^x
6=7^x
log6=xlog7
x=log6/log7
x≈.9208..

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In an isosceles triangle, the base is 7 more than one-half times the legs. If perimeter is 22 centimeters. Find length of each leg.
**
let x=length of each leg
base+2 legs=perimeter
(1/2)x+7+2x=22
x/2+7+2x=22
5x/2=15
5x=30
x=6
length of each leg=6 cm

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write the following as a single logarithm with coefficient of 1. please show steps. thanks in adavance
3logc(x^4)-2logc(3x)
logc[(x^4)^3/(3x)^2]
logc[(x^12/(9x^2]
logc[(x^10/9)]

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Two cars leave towns 640 kilometers apart at the same time and travel toward each other. One car's rate is 10 kilometers per hour more than the other's. If they meet in 4 hours, what is the rate of the faster car?
**
let x=rate of speed of faster car
x-10=rate of speed of slower car
distance=rate of speed*travel time
4x+4(x-10)=640
4x+4x-40=640
8x=680
x=85
rate of speed of faster car= 85km/hr

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A metal alloy weighing 12 lb. and containing
20% copper is melted and mixed with 3 lb. of a
different alloy which contains 40% copper.
What percent of the resulting alloy is copper?
**
20%*12+40%*3/12+3=(2.4+1.2)/15=3.6/15=.24
What percent of the resulting alloy is copper? 24%

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Solve the logarithmic equation algebraically.Approximate the result to three decimal places.
log(4x)-log(14+the square root of x)=2
**
log(4x)-log(14+√x)=2
log[4x/(14+x^(1/2))=2
convert to exponential form
10^2=4x/(14+x^(1/2))=100
4x=1400+100x^(1/2)
4x-1400-100x^(1/2)=0
let u=x^(1/2)
u^2=x
4u^2-100u-1400=0
u^2-25u-350=0
(u-35)(u+10)
u=35=√x
x=35^2=1225
or
u=-10=√x
x=(-10)^2=100
ans:
x=1225
or
x=100

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Solve each equation for x
A). ln(2x-3)=-1
2x-3=e^-1=1/e
2x=1/e+3
x=(1/e+3)/2
x=1.6839
..
B). ln(3x+1)=2
3x+1=e^2
3x=e^2-1
x=(e^2-1)/3
x=2.1297

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Solve each exponential equation for x. Give an exact solution and also approximate the solution to four decimals
a). 4^(3x+2)=9
(3x+2)log4=log9
(3x+2)=log9/log4
3x=(log9/log4)-2
x=[(log9/log4)-2]/3 (exact solution)
x=-0.1383..
..
b). 5^(3x-5)=4
(3x-5)log5=log4
3x-5=log4/log5
x=[(log4/log5)+5]/3 (exact solution)
x=1.9538..

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Find all the real number roots of the equation.
log2(x+7)-log2(x-3)=3
log2[(x+7)/(x-3)]=3
convert to exponential form
(x+7)/(x-3)=2^3=8
x+7=8x-24
7x=31
x=31/7=4.4286..

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log5(x-2)= 2 + log5(x-4)
log5(x-2)- log5(x-4)=2
log5[(x-2)/(x-4)]=2
convert to exponential form
(x-2)/(x-4)=5^2=25
x-2=25x-100
24x=98
x=4.0833..

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Solve each exponential equation for x. Give an exact solution and also approximate the solution to four decimals
2*5^(x-1)=1
log2+(x-1)log5=log1
log1=0
log2+(x-1)log5=0
log2+(x-1)log5=-log2
x-1=-log2/log5
x=(-log2/log5)+1 (exact solution)
x=0.5963..

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randy invested his inheritance in an account that paid 6.9% interest, coupounded continuously. after 5 years, he found that he now had $52,565.56. what was the orginal amount of his inheritance?
**
Formula for continuous compounding:A= Pe^rt, P=initial investment, r=rate of interest, A=amt after t years
52565.56=Pe^(.069*5)=Pe^.345
P=52565.56/e^.345=37228
orginal amount of his inheritance=$37,228.00..

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E is the ellipse with foci at (4,-2) and (4,8) and whose major axis has length of 20
Find an equation for the indicated conic section
**
This is an ellipse with vertical major axis.
Its standard form of equation:, a>b, (h,k)=(x,y) coordinates of center
For given ellipse:
center: (4,3)
given length of vertical major axis=20=2a
a=10
a^2=100
c=5 (distance from center to foci)
c^2=25
c^2=a^2-b^2
b^2=a^2-c^2=100-25=75
Equation of given ellipse:

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Find the exact values of each trigonometric function at 0(theta)=-(7pi/12). For sin cos and tan.
-(7π/12) is in quadrant III where sin<0, cos<0, tan>0
-(7π/12)=[(-3π/12)+(-4π/12)]=[(-π/4)+(-π/3)]
sin-(7π/12)=sin[(-π/4)+(-π/3)]=[sin(-π/4)cos(-π/3)]+[cos(-π/4)sin(-π/3)]
=[(-√2/2)*(1/2)+√2/2*-√3/2]
=[-√2/4-√6/4]
=(-√2-√6)/4
sin-(7π/12)=-(√2+√6)/4
check with calculator:
sin-(7π/12)=-0.9659..
-(√2+√6)/4=-0.9659..
..
cos-(7π/12)=cos[(-π/4)+(-π/3)]=[cos(-π/4)cos(-π/3)]-[sin(-π/4)sin(-π/3)]
=[(√2/2)*(1/2)]-[-√2/2*-√3/2]
=[√2/4-√6/4]
=(√2-√6)/4
check with calculator:
cos-(7π/12)=-0.2588..
(√2-√6)/4=-0.2588..
..
tan-(7π/12)=tan[(-3π/12)+(-4π/12)]=tan[(-π/4)+(-π/3)]
=[tan(-π/4)+tan(-π/3)]/[1-tan(-π/4)*tan(-π/3)]=[-1+(-√3)]/[1-(-1*(-√3)]
=(-1-√3)/(1-√3)
check with calculator:
tan-(7π/12)=3.732..
(-1-√3)/(1-√3)=3.732..

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Find the six trig functions of -pi
sin -π=0
cos -π=-1
tan -π=0
csc -π=undefined
sec -π=-1
cot -π=undefined

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Given that sin(3θ)= (1/5) and (π/2)<3θ<π: Find cos(3θ), sin(6θ), and tan (3θ/2)
You are working with reference angle 3θ in quadrant II where sin>0, cos<0, and tan<0
sin3θ=opposite side/hypotenuse=1/5
adjacent side=√(5^2-1^2)=√(25-1)=√24
..
cos(3θ)=-√24/5
..
Identity: sin 2s=2sins cos s
sin(6θ)=2sin3θcos3θ
=2*1/5*-√24/5
=-(2√24)/25
..
Identity: tan s/2= (sin s)/(1+cos s)
tan (3θ/2)=sin3θ/1+cos3θ
=(1/5)/(1-√24/5)
=(1/5)/(5-√24)/5
=1/(5-√24)

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Solve each equation on the interval (0,2pi): a.) 2sin^(2)x-cos x-1=0, b.) sin 3x cos 2x + cos3x sin2x= (√2/2).
**
a.) 2sin^(2)x-cosx-1=0
2(1-cos^2x)-cosx-1=0
2-2cos^2x-cosx-1=0
2cos^2x+cosx-1=0
(2cosx-1)(cosx+1)=0
2cosx-1=0
cosx=1/2
x=π/3, 5π/3
or
cosx+1=0
cosx=-1
x=π
..
b.) sin 3x cos 2x + cos3x sin2x= (√2/2).
Identity:sin(s+t)=sin s cos t+cos s sin t
sin 3x cos 2x + cos3x sin2x= (√2/2)
sin(3x+2x)=sin(5x)=√2/2
5x=π/4, 3π/4
x=π/20, 3π/20

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Solve for t
10^(7t) = 18^(2t-5)
7tlog10=(2t-5)log18
7t/2t-5=log18/log10≈1.2553
7t=1.2553(2t-5)=2.5105t-6.2764
7t-2.5105t=-6.2764
4.4894t=-6.2764
t≈-6.2764/4.4894
t≈-1.3980

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The inverse of the log function is the exponential form:
For given problem:y=log 9 (x)
Exponential form: base(9) raised to log of number(y)=number(x)
9^y=x
so, you were right on.

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Log2t^3-Logt = Log16+Logt
log[2t^3/t]=log[16/t]
2t^3/t=16/t
2t^3=16
t^3=8
t=2

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Write as a single logarithm:
1/3[logsubscript4x-logsubscript4y] + 2logsubscript4(3x-4)
1/3[log4(x)-log4(y)]+2log4(3x-4)
log4[(x/y)^(1/3)*(3x-4)^2]

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Write the expression as the sine, cosine, or tangent of an angle.
sin 48° cos 15° - cos 48° sin 15°
Identity: sin(s-t)=sin s cos t-cos s sin t
sin 48° cos 15° - cos 48° sin 15°=sin(48º-15º)=sin 33º
..
And also:
Find all solutions to the equation.
cos x = sin x
tan x=cos x/sin x=1
x=45º+360ºk, x=225º+360ºk, k=integer
..
And:
Verify the identity.
4 csc(2x) = 2 csc^2x tan x
start with right side
2csc^2x tanx
=2/sin^2x*sinx/cosx
=2/sinx cosx
multiply top and bottom by 2
=4/2sinx cosx
=4/sin(2x)
=4csc(2x)
verified: right side=left side

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A man drove 156 km at a constant rate. If he had driven 5km/hr. faster he would made the trip in 12 minutes less time. How fast did he drive?
ans. 60 KPH
**
let x=how fast he drove
x+5=speed in 12 min less time
156/x-156/(x+5)=12 min=12/60 hr=0.2 hr
156/x-156/(x+5)=.2
LCD: x(x+5)
156(x+5)+156x=x(x+5)(.2)
156x+780+156x=.2x^2+x
.2x^2+x-780=0
2x+10x-7800=0
x+5x-3900=0
(x+65)(x-60)=0
x+65=0
x=-65(reject, x>0)
or
x-60=0
x=60
how fast he drove=60 km/hr

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log(2x)=1.7
Solve for x
2x=10^1.7
x=(10^1.7)/2
x≈25.0594

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2. Solve the following exponential equation. Exact answers only.
49^x-12*7^x=-36
7^2x-12*7^x+36=0
let u=7^x
u^2=7^2x
u^2-12u+36=0
(u-6)(u-6)=0
u=6=7^x
xlog7=log6
x=log6/log7