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IF N= -3 AND X =(N_5), WHAT IS THE VALUE OF X?
it is assumed that X =(N_5), is a typing mistake and it really means N-5..
we have to put N=-3 in the the given equation for X=N-5=-3-5=-8

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f(x) = 3x^2-x+7, f(1)
f(x) on the left hand side of equal to means function of x where x is an independent variable which can take different values and the f(x) given on the right hand side of equal to(a polynomial in x in this case),depends on the x value chosen in that we have to simply substitute the value chosen for x in the f(x),wherever x occurs.so f(a) means ,put a in place of x wherever x occurs in the right hand side.
hence to find f(1),we have to substitute 1 in place of x on the rhs of equation describing f(x)
f(1)=3*1^2-1+7=3*1-1+7=3-1+7=9

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the degree of a polynomial is the value of the highest power in the polynomial.when you add a polynomial of degree m to a polynomial of degree n ,with m>n , the highest power is still m only.so the degree of resultant polynomial is m

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SEE THE EXAMPLE BELOW
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Answer 8366 by venugopalramana(134) on 2005-10-27 01:12:07 (Show Source):
length=x,width= 4x, and height=9x are the 3 sides of the rectangular box
volume = length *width*height=x*4x*9x=36x^3..so degree is 3 and the leading coefficient is 36 and the formula is volume=36x^3

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the degree of a polynomial is the value of the highest power in the polynomial.when you add a polynomial of degree m to a polynomial of degree n ,with m>n , the highest power is still m only.so the degree of resultant polynomial is m

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the product of two consecutive positive even integers is twelve more than three times the smaller integers find the integers.
let the smaller even number be 2n and next even number is 2n+2..
their product=2n(2n+2)=4n^2+4n..this is equal to ...
three times smaller integer+12=3*2n+12=6n+12..hence
4n^2+4n=6n+12
4n^2-2n-12=0
2n^2-n-6=0
2n^2-4n+3n-6=0
2n(n-2)+3(n-2)=0
(n-2)(2n+3)=0
hence n=2 is the smaller integer and 4 is the next integer....
plus the base of a triangle is 2cm less than four times the height. the area of thr triangle is 45 sq cm. find the height and the length of the base of the triangle.
area of a triangle=(1/2)*base*height
let the height be h..four times the height =4h...less 2 cm =4h-2 ..this is equal to base =b
hence b=4h-2
area=45=(1/2)(4h-2)h=(2h-1)h=2h^2-h
2h^2-h-45=0
2h^2-10h+9h-45=0
2h(h-5)+9(h-5)=0
(h-5)(2h+9)=0..
hence h=5
base 4h-2=4*5-2=18

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a.)8x-4y=16 slope intercept form of a st.line is Y=mX+c..so rewrite the eqn.as required..
8x-16=4y
4y=8x-16
y=(8/4)x-(16/4)
y=2x-4..which is in the slope intercept form..with slope-m=2 and intercept=c=-4
NOW YOU CAN DO THE OTHER PROBLEMS IN THE SAME WAY.

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3a+b+c=7..........(1)
a+3b-c=13........(2)
b=2a-1..........(3)...substituting value of b from eqn.(3) in eqns.(1)and(2)
we get...3a+2a-1+c=7...or...5a+c=7+1=8..............(4)
a+3*(2a-1)-c=13...or...a+6a-3-c=13...or...7a-c=13+3=16.........(5)
eqn.(4)+eqn.(5)...12a=8+16=24..or..a=24/12=2....substituting in (3)..we get
b=2*2-1=3...substituting in (4)we get..5*2+c=8..or..c=8-10=-2..
so a=2;b=3;c=-2.

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4(2x-3)-3(3x-10)=11
8x-12-9x+30=11
-x=11+12-30=-7
x=-7/-1=7
x=7

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x/3 + x/4 = 14
x(1/3+1/4)=14
x((4+3)/12)=14
x(7/12)=14
x=14*12/7=24
x=24

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9x-y-2z=18...........(1)
5x-2y+z=10...........(2)
4x+y-3z=8 ...........(3) ....we find that eqn(2)+eqn(3)gives us 9x-y-2z=18 which is same as (1)..so they are not independent equations .they are dependent equations and consistent in the sense that there are no contradictions.hence we get infinite solution for such a set of equations.we can give any value we like for one variable and solve the rest with that..say let us take x=0...so we get
-y-2z=18..........from (1)
y-3z=8.............from(3)
adding the above -5z=26..or...z=-26/5..substituting...
-y-2*(-26/5)=18
-y=18-52/5=(90-52)/5 = 38/5..or y=-38/5....so one solution set is
x=0;y=-38/5;and z=-26/5
like this we can get any number of solution sets by giving different vaslues to one variable and solving for the rest

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3(x+2)-2(3x-5)=10
3x+6-6x+10=10
-3x=10-10-6=-6
x=-6/-3=2

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we need a system of eqns.,that has 2,7.5 as its solution.there are 2 solutions,hence we need to have 2 variables..let them be x and y
let us find their sum =x+y=2+7.5=9.5
let us find their difference=x-y=2-7.5=-5.5
hence the required system of eqns.are
x+y=9.5 and x-y = -5.5

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what is the leading coefficient for p(x)=(x+2)^17..
if you mean by leading coefficient the coefficient of highest power of x , then notice that
p(x)=(x+2)^17=(x+2)(x+2)(x+2).....17 times
the highest power of x is obviously got by multiplying x by x repeatedly taking one x from each factor ..so we get
x*x*x*.....17 times =x^17...its coeffiocient is 1 and the term is 1x^17

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SEE THE FOLLOWING EXAMPLE AND DO IT IN A SIMILAR WAY
SOLVE 3(2x-1)=5(3x-15)
3*2x-3*1=5*3x-5*15
6x-3=15x-75
15x-75=6x-3
15x-6x=-3+75
9x=72
x=72/9=8

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3(2x-1)=5(3x-15)
3*2x-3*1=5*3x-5*15
6x-3=15x-75
15x-75=6x-3
15x-6x=-3+75
9x=72
x=72/9=8

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10 to the -5 power =0.00001
use the formula x^(-a)=1/x^a...we get
10^(-5) = 1/10^5=1/(10*10*10*10*10)=0.00001

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y=3/5x + 1/4 intersects the line y=-1/8x + r at the point (-1/2,-1/20)
the intersection point -(1/2,-1/20) lies on both the lines.hence
y=-1/8 x + r should be satisfied by (-1/2,-1/20)..substituting
-1/20=(-1/8)*(-1/2)+r
r=-1/20-(1/16) = -[(4+5)/80]=-9/80

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g(x)= -2x^2-8x+12 =0 or less than zero
first solve g(x)=0 using


= are the 2 solutions in the case when g(x)=0
For the case g(x)<0 we have
g(x)= [x-(-2+10^0.5)][x-(-2-10^0.5)]<0..product of 2 factors will be negative if one of the 2 is negative and the other positive
hence g(x)<0 if x lies between (-2+) and (-2-)

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i)^42.....is it (-1)?
we know that
i=i
i^2=-1
i^3=i^2*i=-1*i=-i
i^4=(i^2)^2=1...etc
so we see that if the power is even number then the answer is 1 or -1 and if power is odd number then the answer is i or -i
now in the problem i^42 we know that 42 is even number .so answer is 1 or -1 ..now to find whether it is +1 or -1 ..we have to find it is even number divisible by 4 or not. if it is divisible by 4 the answer is +1 other wise the answer is -1.
here 42 is not divisible by 4 ..so the answer is -1

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Martina leaves home at 9 A.M., bicycling at a rate of 24mi/h. Two hours later, John leaves, driving at a rate of 48mi/h. At what time will JOhn catch up with Martina?

I have been trying to figure this out for several hours now. Is this a trick question, because I cannot figure it out. I know that distance= rate times time (d=r x t), but it doesn't give the distance. this is OK ..but you have to learn the concept of RELATIVE SPEED HERE to do the problem.let me explain the principle first before we solve your problem
let us say there are 2 persons ..you and your friend martina..let us say you are at one place to start with at your home and both of you start at the same time.you walk at 4 mph(miles per hour)say and martina walks at 3 mph.now see what happens after 1 hr.using your formula d=r*t,you will be 4 miles away from your home and martina will be 3 miles away...after 2 hours the distances will be 8 and 6 miles respectively...like this though you started at the same time and at the same place that is zero distance between you and martina,your distance of seperation increased to 4-3=1 mile in 1 hour ,8-6=2 miles in 2 hrs...and so on..so now you are ready to take this concept of relative speed ...relative speed between you and martina is 4 mph - 3 mph = 1mph.in this case as you are travelling in the same direction.so you shoud use this relative speed for r in your formula of d=r*t...ok..the d in this case is the distance of seperation between you and martina at the start .it may be zero as we assumed in this case or may not be zero as given in your problem.similarly you mave to adjust for difference in time of start...but the principle is same so procedure is
1.use relative speed.. this will be difference in speeds if you are travelling in same direction.BUT IT WILL BE SUM OF THE SPEEDS IF YOU ARE TRAVELLING IN OPPOSITE DIRECTIONS.CAN YOU GUESS WHY?BECAUSE AS WE ANALYSED ABOVE EVERY HOUR YOU TWO ARE APPROACHING EACH OTHER (HERE OFCOURSE AT THE BEGINING YOU TWO WILL HAVE TO BE SEPERATED BY A CERTAIN DISTANCE)AND THE DISTANCE OF SEPERATION REDUCES BY THE SUM OF YOUR SPEEDS EVERY HOUR.
2.find the difference in time of start if any and its effect on distance of seperation at the start.
3find the distance of seperation at start using given data and time gap mentioned in 2.
4.note the end criteria..and use your standard formula d=r*t to meet the end criteria using relative speed for r and difference in distance of seperation between start and end positions.
now let us see your example...
1.relative speed =48-24=24 mph=r
2.difference in time of start =2 hrs.
3.distance of seperation at the start = the head start martina got in 2 hrs cycling at 24 mph=2*24=48 miles..
4.end criteria is that they meet at the end ..that is distance of seperation at the end is zero.so difference in distance of seperation from start to end ..... =48-0=48=d
so using d=r*t,we get 48=24*t..or..t=48/24=2 hrs. that is john will catch up with martina in 2 hrs from his time of start that is 11=00+2....that is 13=00 hrs.

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length=x,width= 4x, and height=9x are the 3 sides of the rectangular box
volume = length *width*height=x*4x*9x=36x^3..so degree is 3 and the leading coefficient is 36 and the formula is volume=36x^3

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this is the sum of n consecutive natural numbers from 1 to n .they are in arithmatic progression called a.p the derivation for sum of a.p is done by this simple method of inversion
say S(100)=1+2+3......+98+99+100....reverse this order of summation
S(100)=100+99+98+....3+2+1
adding 2*S(100)=101+101+101+........101+101+101......100 times
2S(100)=101*100
S(100)=101*100/2=5050...is the method...similarly for n numbers we get
S(n)=1+2+3......+(n-2)+(n-1)+(n)....reverse this order of summation
S(n)=(n)+(n-1)+(n-2)+....3+2+1...adding
2*S(n)=(n+1)+(n+1)+(n+1)........(n+1)+(n+1)+(n+1).....n times
2S(n)=n(n+1)
S(n)=(n^2+n)/2=(1/2)*n^2+(1/2)*n+0

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x + 8 = - 42+6x
-42+6x=x+8
6x-x=8+42=50
5x=50
x=50/5=10

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x + 8 = - 42+6x
-42+6x=x+8
6x-x=8+42=50
5x=50
x=50/5=10

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Let: U=-8+9i, v=4+3i, w=6-4i
Solve the following equation for : x
ux+v=w so you do x=w-v/u....NO..NO.dont do this way..proceed as follows..firstly your problem needs a small change AS THE PROBLEM YOU MENTIONED HAS NO SOLUTION.(IF YOU SOLVE IT BY THE METHOD GIVEN YOU WILL FIND THAT YOU GET 2 DIFFERENT ANSWERS FOR x MEANING THAT THERE IS NO SOLUTION SATIFYING THE GIVEN EQUATION.DO IT AND CHECK FOR YOUR SELF).hence take it as ux+vy=w
x(-8+9i)+y(4+3i)=(6-4i)
(-8x+4y)+i(9x+3y)=6-4i...the real part on the lhs of the eqn.should equal the real part on rhs of the eqn.same applies to imaginary parts...hence
-8x+4y=6..and 9x+3y=-4...now i think you can solve the 2 simultaneous eqns.

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y+3x= -5......(1)
2y-3x= 9.....(2)
add eqn(1) and eqn(2)..we get
y+3x+2y-3x=-5+9=4
3y=4
y=4/3
substitute in eqn.1...4/3+3x=-5
3x=-5-4/3=(-3*5-4)/3=-19/3
x=-19/(3*3)=-19/9

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a father is now 24 years older than his son. In 8 years the father will be twice as old as his son will be then. Find there present ages
let the son's age now be s
24 years older than this =24+s=age of father now
in 8 years son will be 8+s...twice as old as this =2*(8+s)=16+2s....(1)
in 8 years father will be 24+s+8=32+s.......(2)
it is given that (1)=(2).hence
16+2s=32+s
2s-s=32-16=16
s=16...hence son's present age is 16 years and father's present age is 24+16=40 years.

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a father is now 24 years older than his son. In 8 years the father will be twice as old as his son will be then. Find there present ages
let the son's age now be s
24 years older than this =24+s=age of father now
in 8 years son will be 8+s...twice as old as this =2*(8+s)=16+2s....(1)
in 8 years father will be 24+s+8=32+s.......(2)
it is given that (1)=(2).hence
16+2s=32+s
2s-s=32-16=16
s=16...hence son's present age is 16 years and father's present age is 24+16=40 years.

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5sqrt72-3sqrt48-4sqrt128=?...the procedure is to factorise the radicals and take as much as possible out side the radical and simplify as per the classification
for ex...sqrt72=sqrt2*2*2*3*3..there are three 2's and two 3's .so we can take two each of them out side the radical of square root,leaving only one 2 in the radical...
sqrt72=sqrt2*2*2*3*3=2*3sqrt2=6sqrt2...similarly
sqrt48=sqrt2*2*2*2*3=2*2sqrt3=4sqrt3
sqrt128=sqrt2*2*2*2*2*2*2=2*2*2sqrt2=8sqrt2..
now you should know that in their simplest form like sqrt2,sqrt3,sqrt5...etc can not be simplified any further and can be added or subtracted within their category only..
so we have now 5sqrt72-3sqrt48-4sqrt128=5*6sqrt2-3*4sqrt3-4*8sqrt2=sqrt2(30-32)-12sqrt3=-2sqrt2-12sqrt3...answer

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9+3x=27 ..to solve bring all unknowns to one side and knowns to the other side of equal to ,simplify and get the answer
3x=27-9=18
x=18/3=6
hence x=6...for a check substitute 6 for x in the given equation and verify.
9+3*6=9+18=27=right hand side ...ok