±√324 - how do you solve this

A method of extracting roots is to use the factorization of a number (by successively dividing out primes until the number is manageable) 
factoring 324... 
324=2*162=
2^2*81=2^2*9^2 

so that 324 = (2*9)^2 = 18^2 

therefore the square root of 324 is 18 
±√324 = ±18

:)

cos2x-cos6x=0 find the exact values of all solutions between 0 and pie

 
 
so 
 
then for  
we get 
,, 

A sampling grew 5 1/2 inches one year 6 3/4 inches the second year,and 7 1/4 inches the third year.how much did it grow during the three years

Essentially we are adding these three mixed numbers: 

5 1/2 + 6 3/4 + 7 1/4 

the common denominator is 4, so we convert: 

5 2/4 + 6 3/4 + 7 1/4 = 
5+6+7 + 2/4+3/4+1/4 = 
18 6/4= 
18 + 4/4 +2/4= 
19 2/4= 
19 1/2

7 over 2 x + 1 over 2 x = 3x + 3 over 2 + 5 over 2 x

I believe you are describing: 
 
so that... 




 
 
 

leads to 

 
which has solutions: 

Simplify: 4x to the 4th power/2x to the 3rd power

1/2 (log7 x + log7 8) = log7 16

1/2 (log7 x + log7 8) = log7 16 
 

 

raising 7 to both sides yields: 
 
i.e. 
 

so that 
 

:)

jane has 43 cents. she has 8 coins. find the coins and the number of each that jane has.

clearly she had 3 pennies (why?) 
that leaves 5 coins to make 40 cent.
quarters aren't can't work...(why?)
...so how about 3 dimes and 2 nickels? 

:)

two complementary angles are in the ratio 3:2

1) x/y = 3/2 
or (2/3)x=y 

2) x+y=90 

together 
x + (2/3)x=90 
5/3 x = 90 

(3/5)(5/3)x=(3/5)*90 
x=54 
thus y=36 

:)

if 5 miles is equal to 8km how many miles is 120 km

This is a proportion: 
5/8 = x/120 
x= 120*5/8=75 
75 mi 

:)

Range of sinx + cosx + 3

Although the ranges of sine and cosine are both 1, together their magnitude never reaches 2: 





Range:

evaluate tan(sin^-1 (p/sqrt p^(2)+9))

Assuming you meant to write: 
 
t=p/sqrt (p^(2)+9))
note that sin(t)=p/r 
and r=sqrt (p^(2)+9) 
from which we see that x=3, y=p 
and so 
tan(t) = p/3 

A rectangular pool is 21 meters wide and 28 meters long. If you swim diangonally across the pool, how many meters would you be swimming?

21-28-35 are in the 3-4-5 family of right triangles so 35 meters

assume that SAT scores are normally distributed with mean u=1518 and standard deviation of 325 B) If 100 SAT scores are selected find the probability that they have a mean less than 1500

n=100
u=ux=1518
o=325
ox=325/root(100)=32.5 


P(x<1500)=
P((x-ux)/ox < (1500-1518)/32.5)=
P(z<-0.5538)=
0.2898

A 180-pound man went on a diet and lost 8 3/4 pounds the first month and 7 1/2 pounds the second month what was his weight after months? (Hint: Add the two losses and then .subtract from starting)

180 - 8 3/4 - 7 1/2 = 
180-8-7-3/4 -2/4 = 
165 - 5/4=
165 - 1 1/4= 
163 3/4

:)

add: 4 1/2+ 3 1/3 +2 7/8

we use the lcd 24 to write

4 12/24 + 3 8/24 + 2 21/24

then we add...
9 41/24

then we change improper fraction(s) to proper...
10 17/24

:)

Find two consecutive odd integers such that 67 more than the lesser is six times the greater. The lesser consecutive off integer is __, and the greater consecutive odd integer is __.

n+67 = 6(n+2) 
5n=67-12
n=11 
they are 11 and 13

If tan α = 9 and α is acute, find the exact value of sin 2α.

we use a for alpha... 

tan(a)=sin(a)/cos(a)=9 
and 
tan(a)=9=9/1 
and thus y=9 , x=1, r=sqrt(82) 
so that 
sin 2a = 
2sin(a)*cos(a)=
2(y/r)(x/r)=
2(9)(1)/82=9/41

The mean, the median, and the mode of the five numbers below are all equal. What number does A represent? 1.8 1.6 2.1 1.7 A

sorted: 1.6, 1.7, 1.8, 2.1 


Mean = (A+7.2)/5 


the middle choices are 1.7 and 1.8. Test 1.7: 

A=5*1.7-7.2= 1.3 (no because this value leaves us without a mode) 

Testing 1.8: 

A=5*1.8-7.2= 1.8 


A=1.8 

:)

Find the indicated probability If P(A or B) = 0.7, P(A) = 0.3, and P(A and B) = 0.15, find P(B) Thank you

P(A or B) = P(A) + P(B) - P(A and B) 

P(B) = P(A and B)+ P(A or B)-P(A)= 


0.15 + 0.70 - 0.3 = 


0.55 


:)

One number is 3 times another. If the sum of their reciprocals is 2/9, find the two numbers.

y = 3x 
1/x + 1/y = 2/9 
or 
1/x + 1/(3x) = 2/9 
3/(3x)+1/(3x) = 2/9 
4/(3x) = 2/9 
taking reciprocals: 
(3x)/4 = 9/2 
(4/3)((3x)/4) = (4/3)(9/2) 
x=6 

:)

One-fourth of the sum of three consecutive integers is 12 less than the smaller integer

n - least number 
 
then we can write 
 

 

so that n=51

Parallelograms ABCD is placed on a coordinate grid. A (5, -1), B (7, 4), and C (7, 8). Find the coordinates for D.

(5,3) because points B and C are vertical to each other (so D is vertical to A) 

:)

how long will it take for the population of a certain country to quadruple if its annual growth rate is 2.5% round to the nearest year.

Growth model: 
 
which we simplify to get 

 
solving for t ... 
 


 
this is about 81 years

Find the quadratic equation whose roots are twice the roots of 4x^2+8x-5=0.

ax^2+bx+c
4x^2+8x-5=0 
since a*c=-20 
we use 10 and -2 as follows 

4x^2+10x-2x-5=0 
2x(2x+5)-1(2x+5)=0 
(2x-1)(2x+5)=0 
so x=1/2 or x=-5/2 
and so 
(x-1)(x+5)=0 
x^2-6x-5=0 

:)

In experiment of tossing a die what is the probability of face having 3 dots or 6 dots to appear

These are mutually exclusive so
P(A or B) = P(A)+P(B)=1/6+1/6=1/3

If f(x)=9x^3+4x^2-x+c and f(2)=1 what is the value of c

f(2)=1
implies 
9(2)^3+4(2)^2-2+c=1 
72+16-2+c=1 
c=-85 

:)

(fog)(x) for f(x)=2x-3 and g(x)=x^2 - 3x - 2 how do I work this out?

(fog)(x) = 
f(g(x))= 
2g(x)-3= 
2(x^2 - 3x - 2)-3= 
2x^2 - 6x - 7 

:)

Determine the sign of cos pi divided by seven. and Is the function cot t positive or negative in Quadrant II?

1) pi/7 is in quad I and all trig functions are positive there, so positive 

2) cot is negative in II

cos^4a+cos^2a=1 ,prove is it identity or not

not an identity:
To disprove an identity we need only a single exception: 
 
and not 1 

:)

sec^4 a+sec^2 a=1 ,prove is it identity or not,thanks and regards

clearly not an identity try a=pi, substitution leads to 
2=1 
:)

(a+5b-4)^3

(a+5b-4)^3= 


(a+5b-4)(a+5b-4)(a+5b-4)= 


(a+5b-4)[a(a+5b-4)+5b(a+5b-4)-4(a+5b-4)]= 

(a+5b-4)[a^2+5ab-4a+5ab+25b^2-20b-4a-20b+16]= 

(a+5b-4)[a^2+10ab+25b^2-8a-40b+16]= 
a[a^2+10ab+25b^2-8a-40b+16]+
5b[a^2+10ab+25b^2-8a-40b+16]-
4[a^2+10ab+25b^2-8a-40b+16]= 


a^3+15a^2b-12a^2+75ab^2-120ab+48a+125b^3-300b^2+240b-64

solve 7x^2+10x=8

7x^2+10x=8
7x^2+10x-8=0
7x^2+14x-4x-8=0
7x(x+2)-4(x+2)=0
(7x-4)(x+2)=0 
x=-2 or x=4/7 

:)

A 6 character computer password is made up of 4 numbers followed by 2 letters. How many different passwords are possible?

counting principle applies: 
(10)(10)(10)(10)(26)(26)=6760000 

:)

Can you help me solve for real roots? 2x^8+15x^4=27

u=x^4 
2u^2+15u-27=0 
2u^2+18u-3u-27=0 
2u(u+9)-3(u+9)=0 
(2u-3)(u+9)=0 

but u+9 contributes no real roots 

2u-3=0 
u=3/2 

x^4 = 3/2 

x^2 = +- sqrt(3/2) 

from which we use the principal root 

thus 
 
or 

Derive a quadratic equation using the following points: (1, -4), (2, -5), (3, -10)

y = ax^2+bx+c 
(1) (1, -4) -> -4=a+b+c , 
(2) (2, -5) -> -5= 4a+2b+c, 
(3) (3, -10) -> -10=9a+3b+c 

(2)-(1): 
-5+4=4a-a+2b-b+c-c 
-1=3a+b (or 3a+b=-1) 
(3)-(1): 
-6=8a+2b (or 4a+b=-3) 

taking the difference of these 2 we get 

4a-3a+b-b=-3+1 (or a=-2) 

and thus b=5 
Now from (1):
-4=-2+5+c (so c=-7) 



y=-2x^2+5x+-7 

:) 

x+(x+y)i=3+8i

Real: x=3 
Imaginary: x+y=8, 3+y=8, y=5 

:)

Find the values of x and y, where x and y are real numbers (3x-4y)+(5/3)yi=-6+5i

3x-4y)+(5/3)yi=-6+5i 
Re:
3x-4y=-6 

Im:
5/3 y=5
y=3 

3x-4(3)=-6 
3x=12-6
3x=6
x=2 
:)

find real numbers x and y if: (x+i)(3-iy)= 1+13i

(x+i)(3-iy)= 1+13i 
(3x+y)+(3-xy)i=1+13i 

(1) 3x+y=1, 
(2) 3-xy=13 
(2) 3-xy=13, xy=-10, y=-10/x 

(1) 3x+y=1, 3x+(-10/x)=1, 3x^2-x-10=0, x=2, x=-5/3 

then 
(x,y)=(2,-5) 
or 

(x,y)=(-5/3, 6)

interval does increase y=x^2+4x

vertex is on the y-axis, opens upwards, so it must increase when x>0, i.e.
(0,oo) 
:)

solve: 8x^2-3=2x

8x^2-3=2x 
8x^2-2x-3=0 
8x^2+4x-6x-3=0 
(8x^2+4x)-(6x+3)=0 
4x(2x+1)-3(2x+1)=0 
(4x-3)(2x+1)=0 

x=3/4 or x=-1/2

a=sin(theta), express sec2(theta) as a function of a.

 
 
therefore we can write 

-7u+3v= 
-7<-2,1>+3<3,2>= 
<14,-7>+<9,6>= 
<23,-1> 

yes it is correct :)

1. the sum of real number and an imaginary number is always? 2. the product of a real number and an imaginary number is always? 3. the quotient of a real number and an imaginary number is always?

1 complex

2 imaginary

3 imaginary

A box containing 3 red balls, 5 white balls, and 10 green balls. If a ball is chosen at random, what is the probability that it is either white or green?

P(w or g) = 
P(w) + P(g) =
5/18 + 10/18 = 
15/18

I need help please. I am having terrible difficulty in solving this problem. Information is given about a polynomial f(x) whose coefficients are real numbers. Find the remaining zeros of f. Degree 3, zeros: 2, -6 - i

the remaining zero is -6+i (since all coefficients are real complex root occur in conjugate pairs) 

:)

2|x+1/3|<9 
means 
-9 < 2(x+1/3) < 9 

-9/2 < x+1/3 < 9/2 

-9/2-1/3 < x < 9/2 - 1/3 
-4 5/6 < x < 4 1/6 

Interval:
(-4 5/6, 4 1/6) 

determine the x and y intercepts of the following linear equation 2x+3y=9

2x+3y=9 
x=0 -> 3y=9, y=3 ...so (0,3) 
y=0 -> 2x=9, x=9/2 ...so (9/2,0) 

:)

The operation * is defined on the set of real numbers by a*b=a+b+ab/2. Is the operation associative

we check that (a*b)*c = a*(b*c) 

(a*b)*c = (a+b+ab/2)*c = 
a+b+ab/2 +c + ((a+b+ab/2)c)/2= 



a*(b*c) = a*(b+c+bc/2) = 
a+(b+c+bc/2)+(a((b+c+bc/2))/2 


comparing... 
(a+b+ab/2 +c + ((a+b+ab/2)c)/2) - 
(a+(b+c+bc/2)+(a((b+c+bc/2))/2) 
which is 0 

the operation is associative 

:)

factorize: 25x^3-10xy^2+xy^2

 
 
:)

What is the distance between P(-3, -2) and Q(5, -2)?

the points are on a horizontal line so d=5-(-3)=8 

:)

What is the distance between points P(2, 7) and Q(10, 13)?

 

 

 
d=10 

During the 1998-1999 Little League season, the Tigers played 48 games. They won 28 more games than they lost. How many games did they win that season?

w+l=48
w-l=28 



2w=72 
w=32 

:)

-7x+y=-19 -2x+3y=-19

m3: -7x+y=-19
-2x+3y=-19 
-21x+3y=-57
-2x+3y=-19 

subtracting the equations... 

-2x+21x+3y-3y=-19+57 
19x=38 
x=2 
which leads us to y=5 

:) 

find the area triangle of sides 5cm,6cm,7cm

where the semiperimeter is

 

a=5, b=6, c=7, s=18/2=9 

by substitution and simplification we get 
 


:)

Solve log3 3x - log3 (x-4) = 2

 

 

 

 


 

x=6