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Which equation, in axsquared + bx + c form, has the roots 5 - 2i and 5 + 2i.
:
x = 5 +/- 2i
Subtract 5 from both sides:
x - 5 = +/- (2i)
Square both sides
(x - 5)^2 = 4(i^2)
FOIL the left, i^2 = -1
x^2 - 10x + 25 = 4(-1)
:
x^2 - 10x + 25 = -4
:
x^2 - 10x + 25 + 4 = 0
:
x^2 - 10x + 29 = 0

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Sally sells cookies at cost of 3 for a $1.00 and mini cakes for $5.00 each.
Sally wants to sell 5 baskets that include all two products for $15.00.
How many cakes and cookies does Sally need to make for her baskets?
:
How about each basket contains 2 cakes and 15 cookies
Cost for this:
2 cakes + 15 cookies
2($5) + ($5) = $15
:
Five baskets;
5(2) = 10 cakes
5(15) = 75 cookies

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Five-sixths of a number is -60.
:
Just write an equation for what it says:
x = -60
Mult both sides by 6
5x = -360
x =
x = -72

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Here are the equations for these problems
:
1. It takes jun thrice the number of days it takes Peter to mow a lawn.
When the two boys work together, the job is completed in three days.
How long does it take each boy to do the job alone?
:
Let t = time required for Peter to mow the lawn
then
3t = time required for Jun to mow the lawn
and
the completed job = 1
+ = 1
:
;
2. Two planes leave NAIA at 4 p.m., one traveling east at 850 kph and the
other, west at 750 kph. At what time will they be 4000 km apart?
:
let t = time required (add to 4 PM)
850t + 750t = 4000
:
:
3. An airplane which travels 180 miles per hour leaves the airport 8.5 hours after a ship has sailed. If it overtakes the ship in one hour and 30 minutes, find the rate of the ship.
:
r = speed of the ship
1.5(180) = r(8.5+1.5)
:
:
4. Rolly and Romy are 15 km apart and are walking towards each other. If rolly is walking at rate of __ kph and romy at a rate of 3 kph, in how many hours will they meet?
:
Let t = time required for them to meet
_t + 3t = 15
:
:
5. Albert sets out on a hike. After walking for a while at 6 kph, he discovers that he has forgotten his lunch. A passing bus takes him back home at 30 kph. How far did he walk if he was gone for one hour?
:
let d = dist walked, (also dist by bus)
+ = 1
:
:
6. The length of a rectangle is six more than half its width.
If the perimeter of the rectangle is 18 units, what are the dimensions of the rectangle?
:
x = the width
then
(.5x+6) = the length
2(.5x + 6) + 2x = 18
:
:
7. The second angle of a triangle is 10 degrees more than half the measure of
the first angle. The third angle is 10 degrees less than four times the second angle.
What are the measures of the three angles in the triangle?
:
let x = 1st angle
then
(.5x+10) = 2nd angle
and
4(.5x+10) - 10 = 3rd angle
:
x + (.5x+10) + (4(.5x+10)-10) = 180

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A woman at a point on the shore of a circular lake with radius 2 miles wants
to arrive on the exact opposite side of the lake in the shortest possible time.
She can walk 4mph and row a boat 2mph. How should she proceed?
:
Find the diameter (rowing directly across the lake) 2 * 2 = 4 mi
Rowing time = = 2 hrs; rowing
:
Find half the circumference: .5(4*pi) = 6.283 mi
Walking time = = 1.57 hrs, walking

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A red and green ball are tossed in the air at the same time. The red ball is
given the velocity of 96 feet per second and its height (t) seconds after it
is tossed is -16t^2+96t feet. The green ball is given a velocity of 80t feet
per second and its height (t) seconds after it is tossed is -16t^2+80t feet.
What is the polynomial difference (t) that represent the difference in the
heights of the two balls? How much higher is the read ball 2 seconds after the
balls are tossed? In reality, when does the difference in the heights stop
increasing?
Do I subtract the second polynomial term from the first to get the difference?
:
Yes, indeed, that's what you do:
f(t) = (-16t^2 + 96t) - (-16t^2 + 80t)
Remove bracket, change signs after the negative;
f(t) = -16t^2 + 96t + 16t^2 - 80t
:
f(t) = -16t^2 + 16t^2 + 96t - 80t
:
f(t) = 16t; the difference between the height of the two balls
;
How much higher is the red ball 2 seconds after the balls are tossed?
t = 2
f(t) = 16(2)
f(t) = 32 ft higher is the red ball
:
In reality, when does the difference in the heights stop increasing?
:
When the green ball hits the ground height, h = 0
Find the time when the height of the green ball = 0
-16t^2 + 80t = 0
Factor out -16t
-16t(t - 5) = 0
the solution we want is
t = 5 seconds
:
:
A graph will illustrate this nicely
:

Note the green ball strikes the ground after 5 sec, while the red ball is still in the air, After 6 sec, both are on the ground and the difference is 0

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solve for x and y in the equations:
x + y =5;
x^y + y^x = 17
:
Just looking at it, it's apparent that: x=2, y=3; or x=3 y=2
:
You could graph it using the formula
y =

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What is the inverse of the function f(x)=x+2/x-2?
:
Write it:
y =
Swap x with y and solve for y:
x =
Multiply both sides by (y-2)
x(y-2) = y + 2
:
xy - 2x = y + 2
:
xy - y = 2x + 2
Factor out y on the left:
y(x-1) = 2x + 2
Divide both sides by (x-1)
y =
:
= ; is the inverse

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Melissa plans to ship a computer in a cubic box that measures x inches on a side.
She later determines that her shipping cost will be less if she uses a
rectangular box that is 3 inches shorter and 5 inches longer than the cubic box.
Write an equation V(x) for the volume of the new box
:
Assuming that the new box has the same height (x)
:
V(x) = x * (x+5) * (x-3)
FOIL
V(x) = x * (x^2 - 3x + 5x - 15)
:
V(x) = x * (x^2 + 2x - 15)
Multiply by x
V(x) = x^3 + 2x^2 - 15x cu/inches

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Mary buys 100 candies for $7.00 .
if yips cost 5 cents each. gups 6 cents each and pips 7 cents each,
How many more pips than yips did mary buy?
:
y + g + p = 100; (total candies)
:
.05y + .06g + .07p = $7.00; (total cost)
:
Multiply the 1st equation by .06 and subtract from the above equation
.05y + .06g + .07p = 7
.06y + .06g + .06p = 6
--------------------------subtraction eliminates g
-.01y + .01p = 1
Multiply above equation by 100 to get rid of the decimals
-1y + 1p = 100
we can write it
p - y = 100, answers the question" How many more pips than yips did mary buy?"

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-3m^2 - 9m = 0
Factor out -3m
-3m(m + 3) = 0
Two solutions
m = 0
m = -3

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Assume the problems is:
= . Find the value of x.
;
=
Cancel 5
= x - 20
Multiply by 3 to get rid of the denominator
x + 3(5) = 3(x - 20)
:
x + 15 = 3x - 60
:
15 + 60 = 3x - x
:
75 = 2x
:
x =
x = 37.5
;
:
You can check solution in the original equation

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A collection of 36 coins consists of nickels, dimes, and quarters.
n + d + q = 36
:
There are three fewer quarters than nickels
n = q + 3
;
and six more dimes than quarters.
d = q + 6
:
How many of each kind of coin are there?
Substitute for n and q in the 1st equation:
(q+3) + (q+6) + q = 36
3q + 9 = 36
3q = 36 - 9
q =
q = 9 quarters
;
I'll let you find the number nickels and dimes

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1) The concentration of salt in Lake Ontario doubles every 7.25 years.
If the current levels are 312.5 parts per million,
a) Find an expression for the concentration of salt in Lake Ontario after n years.
f(n) =
:
b) The concentration levels of salt in the Dead Sea are 10 000 parts per million.
How long will it take for the concentration of salt to reach these levels in Lake Ontario?
:
f(n) = 10000
:
= 10000
:
Divide both sides by 312.5
=
:
= 32
:
32 = 2^5 so we can write it:
= 2^5
therefore
= 5
n = 5 * 7.25
:
n = 36.25 yrs
:
:
Check solution on a calc: enter 312.5 * = 10000

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Two students standing 3000M apart see the same airplane flying in the sky between them.
Student A measures the angle of elevation to be 37 degrees.
Student B measures the angle of elevation to be 30 degrees.
How high is the airplane.
:
Find the 3rd angle of the triangle formed by the plane and the two observers:
180 - 37 - 30 = 113 degrees
:
Find the side (a) opposite the 37 degree angle using the law of sines
+
Find the sines and cross multiply
.9502a = 3000 * .6018
.9502a = 1805.445
a =
a = 1961.364
:
Drop a line from the 113 degree angle and perpendicular to the base,
this is the altitude of the plane
;
This forms two right angle triangles, use the one containing the angle of 30 degrees
Find the altitude (h) using the sine of 30 degrees:
Sin(30) = }
.5 = }
h = .5 * 1961.346
h = 980.673 meters is the height of the plane

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Two cyclists start biking from a trails start 3 hours apart. the second cyclist
travels at 10 miles per hour and starts 3 hours after the first cyclist who is
traveling at 6 miles per hour. How much time will pass before the second
cyclist caches up with the first from the time the second cyclist started biking?
:
Let t = travel time of the 2nd cylist
then
(t+3) = travel time of the 1st cylist
:
When the 2nd catches the 1st, they will have traveled the same distance
Write dist equation: Dist = speed * time
:
2nd cyclist dist = 1st cyclist dist
10t = 6(t+3)
10t = 6t + 18
10t - 6t = 18
4t = 18
t =
t = 4.5 hrs for the 2nd to catch the 1st
:
:
Check solution by finding the distance of each (should be equal)
10 * 4.5 = 45 mi
6 * 7.5 = 45 mi

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Write the equation of the line that contains the points (2, 2) and (-3, 12).
Here is part of the answer. Can you identify the missing part?
y = -2x + ___
Let the missing part = b
y = -2x + b
Substitute for x and y using the 1st given coordinate (2, 2)
2 = -2(2) + b
2 = -4 + b
2 + 4 = b
b = 6
:
the equation y = -2x + 6
:
Check this using the 2nd coordinate (-3, 12)
12 = -2(-3) + 6
12 = +6 + 6

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5x - 3y = 1/3
-10x + 6y = 2/3
:
Multiply the 1st equation by 6, and the 2nd equation by 3, this gets rid of the
denominator and give the coefficient of y the same in both equations
+30x - 18y = 2
-30x + 18y = 2
:
you can see that this has "no solution"

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A piece of wire 30com long is cut into two pieces.
One of these is bent into a circle and the other is bent into a square enclosing the circle.
What is the diameter of the circle?
;
The diameter of the circle is also the side of the square
:
Let x = the diameter and the side of the square
;
Circum + perimeter = 30
pi*x + 4x = 30
;
4.71416x = 30
x =
x = 4.2 cm is the diameter of the circle
:
:
Check
circum: 4.2*pi = 13.195
perime: 4 * 4.2= 16.80
------------------------
wire length = 29.995 ~ 30

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sixty newspapers cost $27.find the cost for 16 newspapers
:
A ratio equation
=
Cross multiply
60x = 16 * 27
60x = 432
x =
x = $7.20 for 16 news papers
:
:
Check by finding the cost of one paper
= .45
Cost of 60
.45 * 60 = $27, confirms our solution

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Let m = man's present age
Let s = son's present age
:
Write an equation for each statement:
:
"ten years ago a man was 3 times as old as his son."
m - 10 = 3(s-10)
m - 10 = 3s - 30
m = 3s - 30 + 10
m = 3s - 20; use for substitution below
:
" in 6 years, he will be twice as old as his son."
m + 6 = 2(s+6)
m + 6 = 2s + 12
m = 2s + 12 - 6
m = 2s + 6
:
how old are they now?
Replace m with (3s - 20) in the above equation
3s - 20 = 2s + 6
3s - 2s = 6 + 20
s = 26 yrs is son's present age
then
m = 2s + 6
m = 2(26) + 6
m = 58 yrs is man's present age
:
:
Check solution in the statement:
"ten years ago a man was 3 times as old as his son. "
58 - 10 = 3(26 - 10)
48 = 3(16)

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rancher removed 200 ft of fence from a field on his ranch.
he wishes to reuse the fencing to create a rectangular corral into which he
will build a 6 ft wide wooden gate.
what dimensions will result in the greatest possible area?
:
Let the side with the gate = (L-6)
then:
L + (L-6) + 2W = 200
:
2L + 2W = 200 + 6
:
2L + 2W = 206
Simplify, divide by 2
L + W = 103
W = (103-L)
:
The Area equation: A = L * W
Replace W with (103-L)
A = L(103-L)
A = -L^2 + 103L
Find the max area by finding the axis of symmetry
L =
L = +51.5, length for max area
Find W:
W = 103 - 51.5
W = 51.5
:
Dimensions for max area: 51.5 by 51.5
:
Check: 51.5 + (51.5-6) + 2(51.5) = 200

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A child in an airport is able to cover ?? meters in ?? minutes running at a steady speed down a moving sidewalk in the direction of the sidewalk's motion. Running at the same speed in the direction opposite to the sidewalk's movement, the child is able to cover ?? meters in ??minutes. What is the child's running speed on a still sidewalk, and what is the speed of the moving sidewalk? pls assist me thank
:
How meters in how many minutes?

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a positive number is 1 more than twice another number.
if the difference of their squares is 40, find the larger of the two numbers.
:
let x = the larger number
let y = the other number
:
"a positive number is 1 more than twice another number."
x = 2y + 1
:
"if the difference of their squares is 40,"
x^2 - y^2 = 40
:
Replace x with (2y+1)
(2y+1)^2 - y^2 = 40
FOIL
4y^2 + 4y + 1 - y^2 = 40
:
3y^2 + 4y + 1 - 40 = 0
:
3y^2 + 4y - 39 = 0
Factor
(3y + 13)(y - 3) = 0
the positive integer solution
y = 3
:
Find x:
x = 2(3) + 1
x = 7
:
Nos. are 7 and 3
:
:
Check: 49 - 9 = 40

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I need to find a quadratic equation for the following table
x y
0 0
2.2 30
2.9 40
4.0 50
5.0 60
6.5 70
8.0 80
9.0 90
11.8 100
:
using the form y = ax^2 + bx + c
x=0, y=0, tells us that c = 0
:
Use two coordinates to find a & b
x = 4, y = 50
x = 9, y = 90
:
x = 4
16a + 4b = 50
:
x = 9
81a + 9b = 90
:
use elimination, mult the 1st equation by 9, the 2nd equation by 4:
144a + 36b = 450
324a + 36b = 360
------------------ subtraction eliminates b
-180a = 90
a =
a = -.5
find b using the 1st equation
16(-.5) + 4b = 50
-8 + 4b = 50
4b = 50 + 8
4b = 58
b =
b = 14.5
:
The equation:
y = -.5x^2 + 14.5x
:
The graph

:
This satifys the data approximately; actual values in brackets:
x y
0 0
2.2 30 (29.5)
2.9 40 (37.8)
4.0 50 (50)
5.0 60 (60)
6.5 70 (73)
8.0 80 (84)
9.0 90 (90)
11.8 100 (101.5


:

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Perform the operation and write the result in standard form.
10i-(3+sqrt(-25))
:
10i-(3+5i)
:
10i - 3 - 5i
:
-3 + 5i

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1. The one's digit of a two-digit number is three more than twice the ten's digit.
the sum of one-third the number and ten is the number itself.
What is the number?
:
Let x = the 10's digit
Let y = the units
then
10x + y = the two digit number
:
Write an equation for each statement:
"1. The one's digit of a two-digit number is three more than twice the ten's digit."
y = 2x + 3
:
"the sum of one-third the number and ten is the number itself."
(10x+y) + 10 = 10x + y
Get rid of the denominator, multiply by 3, results:
10x + y + 3(10) = 3(10x + y)
10x + y + 30 = 30x + 3y
30 = 30x - 10x + 3y - y
30 = 20x + 2y
Substitute (2x+3) for y (from the 1st equation)
30 = 20x + 2(2x+3)
30 = 20x + 4x + 6
30 - 6 = 24x
24 = 24x
x = 1 is the 10's digit
Find y:
y = 2(1) + 3
y = 5
then
15 = the number
:
Check solution in the statement:
"the sum of one-third the number and ten is the number itself."
*15 + 10 = 15
:
:
Do this the same way:
2. The one's digit of a 2-digit number is one more than twice the ten's digit.
if the number is reversed,
the resulting number is two more than twice the original number.
What is the original number?
:
" The one's digit of a 2-digit number is one more than twice the ten's digit."
y = 2x + 1
:
"if the number is reversed, the resulting number is two more than twice the original number."
10y + x = 2(10x+y) + 2
10y + x = 20x + 2y + 2
10y - 2y = 20x - x + 2
8y = 19x + 2
Substitute (2x+1) for y
8(2x+1) = 19x + 2
16x + 8 = 19x + 2
8 - 2 = 19x - 16x
6 = 3x
x =
x = 2 is the 10's digit
find y
You should be able to complete this problem by using the 1st problem as an example

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An aircraft carrier made a trip to Guam and back.
The trip there took three hours and the trip back took four hours.
It averaged 6 km/h on the return trip.
Find the average speed of the trip
:
Find the distance using the time and speed given for the return trip:
d = 4 * 6 = 24 km
:
We know the total traveled distance = 48 km
Given that the total travel time = 7 hrs
:
Av speed =
Av s =
Av s = 6.857 km/hr

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The drama club is selling tickets to a play for $10 each.
The cost to rent the theater and costumes is $500.
In addition the printers are charging $1 per ticket to print the tickets.
How many tickets must the drama club sell to make a profit?
:
Let x = no. of tickets required to make a profit
:
revenue - cost = profit
:
10x - 1x - 500 > 0
:
9x > +500
x >
x > 55.55 tickets ~ 56 need to be sold to make a profit
:
:
Check solution, when 56 tickets are sold
P = 10(56) - 56 - 500
P = 560 - 56 - 500
P = $4 profit

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Debbie traveled by boat 5 miles upstream to fish in her favorite spot.
Because of the 4 mph current, it took her 20 minutes longer to get there
than to return.
How fast will her boat go in still water?
:
Let s = boat speed in still water
then
(s-4) = speed upstream
and
(s+4) = speed down stream
:
Convert 20 min to hrs: 20/60 = hr
:
Write a time equation: time =
Upstream time = downstream time + 20 minutes (1/3 hr)
= +
:
To clear out these denominators, multiply equation by 3(s+4)(s-4)
3(s+4)(s-4)* = 3(s+4)(s-4)* + *3(s+4)(s-4)
Results in
:
15(s+4) = 15(s-4) + (s+4)(s-4)
:
15s + 60 = 15s - 60 + s^2 - 16
:
0 = 15s - 15s - 60 - 60 + s^2 - 16
:
0 = s^2 - 136
:
s^2 = 136
s =
s = 11.66 mph speed in still water
:
:
Check solution on calc: (speed up = 7.66 and speed down = 15.66)
5/7.66 - 5/15.66 = .3333 which is 1/3 of an hr

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:
You are walking across a rail bridge, and are 3/8 of the way across.
You hear a train approaching behind you at 60 miles per hour.
You do some quick calculation, and you realize that you can either run back
or forward, and either way, you can make it to the end of the bridge and get
off the tracks, just in time to avoid getting hit by the train.
That is, you will get to the end of the bridge (either behind you or in front
of you) the exact same time as the train.
What is your running speed?
:
Train----------d------------Br.....3x......Y..........5x.........Br
:
Let d = dist train is from the bridge
Let 8x = length of the bridge
You (Y) are 3x from one end and 5x from the other end of the bridge
:
Let s = running speed (mph)
:
Toward the train time equation (arrive at end of the bridge at the same time)
Train time = your time
=
Cross multiply:
ds = 60(3x)
ds = 180x
d =
:
Away from the train time equation (arrive at other end at the same time)
=
Cross multiply
s(d+8x) = 60(5x)
s(d+8x) = 300x
d + 8x =
Replace d with (from the 1st equation)
+ 8x =
Get rid of x, divide thru by x and you have:
+ 8 =
Multiply by s:
180 + 8s = 300
8s = 300 - 180
8s = 120
s =
s = 15 mph running speed
:
: