Questions on test answered by real tutors!

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Looking at the expression , we can see that the first coefficient is , the second coefficient is , and the last term is .


Now multiply the first coefficient by the last term to get .


Now the question is: what two whole numbers multiply to (the previous product) and add to the second coefficient ?


To find these two numbers, we need to list all of the factors of (the previous product).


Factors of :
1,2,3,4,6,12
-1,-2,-3,-4,-6,-12


Note: list the negative of each factor. This will allow us to find all possible combinations.


These factors pair up and multiply to .
1*(-12)
2*(-6)
3*(-4)
(-1)*(12)
(-2)*(6)
(-3)*(4)

Now let's add up each pair of factors to see if one pair adds to the middle coefficient :

First Number	Second Number	Sum
1	-12	1+(-12)=-11
2	-6	2+(-6)=-4
3	-4	3+(-4)=-1
-1	12	-1+12=11
-2	6	-2+6=4
-3	4	-3+4=1

From the table, we can see that the two numbers and add to (the middle coefficient).


So the two numbers and both multiply to and add to


Now replace the middle term with . Remember, and add to . So this shows us that .


Replace the second term with .


Group the terms into two pairs.


Factor out the GCF from the first group.


Factor out from the second group. The goal of this step is to make the terms in the second parenthesis equal to the terms in the first parenthesis.


Combine like terms. Or factor out the common term

---------------------------------------------


Answer:


So factors to .


Note: you can check the answer by FOILing to get or by graphing the original expression and the answer (the two graphs should be identical).

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solve 6x-4(x+1)= 2(x-3)+2
6x-4x-4 = 2x-6+2
2x-4 = 2x-4
------------
That is true for all values of "x".
The original problem is an identity.
=====================================
Cheers,
Stan H.

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solve 4(x+1)= 2(x-3)-(1-2x)
4x+4 = 2x-6-1+2x
4x+4 = 4x-7
4 = -7
-----------
That is a contradiction which comes about because we assume
all those x's can be replaced by some number. They can't.
The problem has no solution.
-----------------
Cheers,
Stan H.

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4(x+1)=2(x-3)-(1-2x)---->distribute
:
4x+4=2x-6-1+2x---->collect terms on the right side
:
4x+4=4x-7---->which isnt possible!!!!!!

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suppose you roll two dice. find the number of elements in the event space of rolling a sum of 4
------------------------------------
(1,3)/(2,2)/(3,1)
Ans: 3
=============
Cheers,
Stan H.

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These require illustrations and lengthy narrative.
May I suggest you use Google and search for these
terms. You will find many sites that discuss these
topics thoroughly.
Cheers,
Stan H

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1.(2x-3)(x-4)expading this yields
2x^2-8x-3x+12 adding like terms
2x^2-11x+12=0 and you dont have to worry since you were right.
2. x^2+3x-28=2putting likes together we have
x^2+3x-30=0 since this is not a perfect square use the quadratic formula

using the above formula we have
x=4.18 or 7.17
3.x^2+16=-10x putting -10x in the L.H.S we result to
x^2+10x+16=0 solving we get
(x+2)(x+8)=0 hence
x=-2 or -8
4.5x^2=6x-16x^2 grouping like terms
5x^2+16x^2-6x=0
21x^2-6x=0 getting the common term x
x(21x-6)=0
x=0,21x-6=0
21x=6
x=6/21 we therefore have
x=0 or 6/21

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The width of the box is x

The length of the box is (x + 2)

The height of the box is (x - 5)

The volume of a rectangular solid is given by length X width X height, so for this box:



Multiplying:

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a ball is thrown up into the air. its height h, in meters, after t seconds is
h=-4.9t^2+38t+1.75.
for what length of time is the ball above 50 m?
----------------------------------

----------------------------------
-4.9x^2 + 38x + 1.75 = 50
-4.9x^2 + 38x - 48.25 = 0
---
x = [-38 +- sqrt(38^2 - 4*-4.9*-48.25)]/-9.8
x = 1.59973..; x = 6.1554
----------------------------
Total time at or above 50 m: 6.1554-1.59973 = 4.55564.. seconds
===============================================================
Cheers,
Stan H.

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since
A=1/2 b h--------> b=y and h is =x
so A=1/2 (y)(x)
:
as a function of x
:

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4x-2/3=>1/3
4X>=1/3+2/3
4X>=1
X>=1/4 ANS.
PROOF:
LET X=1/3
4*1/3-2/3>=2/3
4/3>=2/3+1/3
4/3>=3/3
1.333>=1

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problem:
x is greater than 1 or less than -2
-----
this would be shown as:
x < -2
or
x > 1
-----
interval notation would look like this:
(,-2) U (1,)
interval line would look like this:
<-------- (- =========== -2) U (1 =========== )----->
that means that x is greater than - infinity and less than -2
OR (the U is an OR symbol)
x is greater than 1 and less than infinity.
-----
the ======= is a fat line which represents the active interval.
the ------- is a skinny line which represents the rest of the interval line not actually part of the active interval.
-----
in interval notation,
(k means x is greater than k
[k means x is greater than or equal to k
k) means x is smaller than k
k] means x is smaller than or equal to k
-----
example:
(-5,10]
means x is greater -5 and less than or equal to 10.
this interval would be shown as -5 < x <= 10
the interval line would look like <----------- (-5 ======== 10] -------->
-----
another example:
[20,100)
means x is greater or equal to 20 and less than 100.
this interval would be shown as 20 <= x < 100.
the interval line would look like <----------- [20 ======== 100) -------->
-----
in your example, the interval line had to be split because the values of x were on the ends of the interval line rather than in the middle.

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x>1 or x<-2

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7-3x<=22...subtract 7 from each side
:
-3x<=15...divide by -3 reverses the inequality sign
:

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A suspension bridge has twin towers that are 1300 feet apart. Each tower extends
180 feet above the road surface. The cables are parabolic in shape and are
suspended from the tops of the towers. The cables touch the road surface at
the center of the bridge. Find the height of the cable at a point 200 feet
from the from the center of the bridge.
:
Find the equation for this:
Three coordinates:
0, 180; the vertical suspension point on the left
650, 0; the center point that touches the road
1300, 180; vertical suspension point on the right
:
Using: ax^2 + bx + c = y
:
0,180, we know c = 180
:
write equation from coordinates;
(650^2)a + 650b + 180 = 0
and
(1300^2)a + 1300b + 180 = 180
:
422500a + 650b + 180 = 0
1690000a + 1300b + 180 = 180
:
Multiply the 1st equation by 2, subtract from the 2nd equation
1690000a + 1300b + 180 = 180
845000a + 1300b + 360 = 0
------------------------------
845000a - 180 = 180
845000a = 180 + 180
845000a = 360
a =
a = .000426

find b:
.000426(650^2) + 650b + 180 = 0
180 + 650b = -180
650b = -180 - 180
b =
b = -.5538
:
The equation; y = .000426x^2 - .5538x + 180
:
Looks something like this:

:
"Find the height of the cable at a point 200 feet
from the from the center of the bridge."
:
200' before the midpoint of 650': x = 450
and
200' after the midpoint of 650': x = 850
:
Substitute these values for x and find y:
y = .000426(450)^2 - .5538(450) + 180
y = 86.265 - 249.21 + 180
y = 17.055 ft
:
You can do the same for x = 850; it is 17.055 ft as you would expect.

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plot the points and the slope of the line passing through the points.
(0,1) (-2,-6)
--------------------
slope = (-6-1)/(-2-0) = -7/-2 = 7/2
--------
intercept: (0,1)
--------
EQUATION:
y = (7/2)x + 1

=====================
Cheers,
Stan H.

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(0,1) (-2,-6)
slope=(y2-y1)(x2-x1)
slope=(-6-1)/-2-0)
slope=-7/-2 or 7/2
1=7/2*0+b
1=b
y=7x/2+1 ans. for the line equation.
(graph 300x200 pixels, x from -6 to 5, y from -10 to 10, 7x/2 +1).

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Find the area of a triangle whose base is 1.3 feet and whose height is 1.5 feet.
------------------------------------------
Area = (1/2)*base*height
A = (1/2)*1.3*1.5
A = 0.975 sq. ft.
====================
Chers,
Stan H.

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414.40=400+400*r*1
414.40=400+400*r
414.40-400=400t
14.40=400t
t=14.40/400
t=.036 or 3.6% is the interest rate.

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Simply isolate W to one side of the equation:
XW + Y = B/4
XW = (B/4) - Y
W = (B/4X) - (Y/X)
.
The only restriction is:
Except when X=0
.
Notice that if X was equal 0, the numerator would be zero and a zero in the denominator would be "undefined".

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please help me with this problem:
Multiply (a+b)(a^2+ab+b^2)
---------------
= a*(a^2+ab+b^2) + b*(a^2+ab+b^2)
= a^3 = a^2b + ab^2 + a^2b + ab^2 + b^3
collect terms
= a^3 + 2a^2b + 2ab^2 + b^3

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You can't have an = inside a radical.

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a picture is 1 inch longer than it is wide. it is put into a frame one half inch wider on every side. find the area of the picture if the frame area is eight inches squared.
:
Let x = width of the picture
then
(x+1) = length of the picture
:
Frame 1/2 inch wider all around, therefore
x + 2(.5)=
(x+1) = width of the frame
and
(x+1) + 2(.5) =
x + 1 + 1 =
(x+2) = length of the the frame
:
Frame area given as 8 sq/in, therefore
(x+1)*(x+2) = 8
FOIL
x^2 + 3x + 2 = 8
x^2 + 3x + 2 - 8 = 0
:
a quadratic equation
x^2 + 3x - 6 = 0
Use the quadratic formula to find x: a=1; b=3; c=-6
:
You should get a positive solution about x = 1.37 inches is the width of the pic
then, 2.37 = length
:
A = 1.37 * 2.37
A = 3.25 sq/in is the picture
:
:
Check solution: Frame dimensions 2.37 by 3.37
Frame area = 2.37 * 3.37 = 7.77 ~ 8 sq/in as given

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84.)Gone Fishing. 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?
-----------------------------
time = distance/speed
time upstream = 5/(b - 4) b is the boat's speed
time downstream = 5/(b+4) = time upstream - 1/3 (20 mins is 1/3 hours)
---------------
5/(b+4) = 5/(b-4) - 1/3
Multiply by 3*(b+4)*(b-4)
15*(b-4) = 15*(b+4) - b^2 + 16
15b-60 = 15b+ 60 -b^2 + 16
-60 = 76 - b^2
b^2 = 136
b = 11.66 mph

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1) 6x^2+24x=2x^2-36
4x^2 +24x + 36 = 0
x^2 + 6x + 9 = 0
(x+3)^2 = 0
x = -3 with multiplicity 2
===================================
2) 3x^2-17x+20=0
3x^2 - 12x - -5x + 20 = 0
3x(x-4)-5(x-4) = 0
(x-4)(3x-5) = 0
x = 4 or x = 5/3
=====================
3) x^2-36=0
(x-6)(x+6) = 0
x = 6 or x = -6
======================
Cheers,
Stan H.

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A = (b*h)/2 (equation 1)
where A = area
b = base
h = height
-----
h = b+6 (given)
A = 13.5 (given)
equation 1 becomes:
A = [b*(b+6)]/2 = 13.5
which becomes:
[b*(b+6)]/2 = 13.5 (equation 2)
-----
multiply both sides of equation 2 by 2:
b*(b+6) = 27
simplify:
b^2 + 6*b = 27
complete the squares:
(b+3)^2 = 27 + 9 (explanation for this down below after the answer)
simplify:
(b+3)^2 = 36
take square root of both sides:
b+3 = +/- 6
subtract 3 from both sides:
b = +/- 6 - 3
b becomes either:
6-3 = 3
or:
-6-3 = -9
-----
since b can't be negative, answer has to be:
b = 3
-----
since h = b+6, then:
h = 9
-----
you have:
b = 3
h = 9
A = 13.5 (given)
-----
substitute in equation 1:
A = (b*h)/2 (equation 1)
A = (3*9)/2 = 13.5
A = 27/2 = 13.5
A = 13.5 = 13.5
-----
values for b and h prove out.
equation is good.
b = 3
h = 9
-----
explanation for completing the squares is shown below:
also shown below is the fact that you could have also solved this using the quadratic formula.
-----
in order to complete the squares of your equation:
b^2 + 6*b)
i took half of the 6 and turned the equation into:
(b+3)^2
if you multiply this out, you will get:
b^2 + 6*b + 9
there's an extra 9 in there.
in order to keep the equations in balance i had to add 9 to the other side of the equation.
that's why the equation went from:
b^2 + 6*b = 27
to:
(b+3)^2 = 27 + 9
the additional 9 on the right hand side of the equation kept them in balance because i added 9 to the left hand side by completing the squares.
-----
use of the quadratic formula instead.
-----
rather than completing the squares, you can also solve this equation using the quadratic formula.
the equation to solve was:
b^2 + 6*b = 27
subtract 27 from both sides of the equation:
b^2 + 6*b - 27 = 0
let x = b (this is necessary because general form of quadratic equation and quadratic formula use b for another purpose)
-----
formula becomes:
x^2 + 6*x - 27 = 0
general form of quadratic equation is:
a*x^2 + b*x + c = 0
the quadratic formula can be used to solve this quadratic equation.
general form of quadratic formula is: x =
a = 1
b = 6
c = -27
-----
b^2 = 36
4*a*c = -108
2*a = 2
-----
quadratic formula becomes
x =
this becomes:
x = (-6 +/- 12)/2
which becomes
x = 6/2 = 3
or:
x = -18/2 = -9
-----
the answers are the same.
you could solve by completing the square or you could solve by quadratic formula, whichever is easiest for you.
-----

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the area of a triangle is 13.5 meters. Find the base and height of the retangle
if its height is 6 meters greater than it's base. Use an equation and the formula area of a triangle =0.5(base)(height).
.
Let b = length of the base
then because "its height is 6 meters greater than it's base"
b+6 = height
.
Plug the above into:
area of a triangle =0.5(base)(height)
13.6 =0.5(b)(b+6)
27.2 =(b)(b+6)
27.2 = b^2 + 6b
0 = b^2 + 6b - 27.2
.
Using the quadratic equation to solve we get:
b = {3.01664358259653, -9.01664358259653}
Throw out the neg solution.
b = 3 meters (base)
b+6 = 9 meters (height)
.
Quadratic solved here:

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Quadratic equation (in our case ) has the following solutons: For these solutions to exist, the discriminant should not be a negative number. First, we need to compute the discriminant : . Discriminant d=144.8 is greater than zero. That means that there are two solutions: . Quadratic expression can be factored: Again, the answer is: 3.01664358259653, -9.01664358259653. Here's your graph:

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Thomas is going to make an open - box by cutting equal squares from th four corners of an 11 inch by 14 inch sheet of cardboard and folding up the sides. If the base is to be 80 square inches, then what size should be cut from each corner?
11x14
base w/corners cut:
(11-2x) by (14-2x)
.
if the base is to be 80 sq inches:
80 = (11-2x)(14-2x)
80 = 154 - 22x -28x + 4x^2
80 = 154 - 50x + 4x^2
80 = 4x^2- 50x + 154
0 = 4x^2- 50x + 74
0 = 2x^2- 25x + 37
Solving using the quadratic equation yields:
x = {10.7845892868043, 1.71541071319574}
.
solution:
10.78 inches
or
1.72 inches
.
Details of quadratic equation:

Solved by pluggable solver: SOLVE quadratic equation with variable

Quadratic equation (in our case ) has the following solutons: For these solutions to exist, the discriminant should not be a negative number. First, we need to compute the discriminant : . Discriminant d=329 is greater than zero. That means that there are two solutions: . Quadratic expression can be factored: Again, the answer is: 10.7845892868043, 1.71541071319574. Here's your graph:

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The number of footballs sold is,

The price for each football is .
The total revenue is just the number of footballs sold multiplied by the price of each football.


.
.
.
When q=$8, then





The weekly revenue when footballs are $8 a piece is $14,400.

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This football problem has been kicking around for few days.
:
If a manufacturer charges q dollars each for footballs, then he can sell 3000 - 150q footballs per week. Find a polynomial R(q) that represents the revenue for one week.
:
Revenue = price times number sold (q=price), (number sold = 3000-150q)
R(q) = q(3000-150q)
;
Multiply what's inside the brackets
R(q) = 3000q - 150q^2
or
R(q) = -150q^2 + 3000q
:
:
Find the weekly revenue if the price is $8 for each football.
:
Take the above equation, substitute 8 for q and find R(q)
R(q) = -150(8^2) + 3000(8)
:
R(q) = -150(64) + 24000
:
R(q) = -9600 + 24000
:
R(q) = $14,400
:
:
Did this explain it to you?

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This football problem has been kicking around for few days.
:
If a manufacturer charges q dollars each for footballs, then he can sell 3000 - 150q footballs per week. Find a polynomial R(q) that represents the revenue for one week.
:
Revenue = price times number sold (q=price), (number sold = 3000-150q)
R(q) = q(3000-150q)
;
Multiply what's inside the brackets
R(q) = 3000q - 150q^2
or
R(q) = -150q^2 + 3000q
:
:
Find the weekly revenue if the price is $8 for each football.
:
Take the above equation, substitute 8 for q and find R(q)
R(q) = -150(8^2) + 3000(8)
:
R(q) = -150(64) + 24000
:
R(q) = -9600 + 24000
:
R(q) = $14,400
:
:
Did this explain it to you?

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The members of a flying club plan to share equally the cost of $200,000 airplane. The members want to find five more people to join the club so that the cost per person will decrease by $2000. How many members are currently in the club?
:
Let x = no. of members currently
then
(x+5) = no. of members to reduce the cost/member $2000
:
Let's deal in thousands of dollars to lessen the number of zeros we have to write:
:
- 2 =
:
Multiply equation by x(x+5) to eliminate the denominators, results
200(x+5) - 2x(x+5) = 200x
:
200x + 1000 - 2x^2 - 10x = 200x
:
Subtract 200x from both sides and we have a quadratic equation
-2x^2 - 10x + 1000
:
Simplify and change the signs, divide equation by -2, results
x^2 + 5x - 200 = 0
:
Factors easily to:
(x+25)(x-20) = 0
;
Positive solution is what we want here:
x = +20 original members
:
:
Check solution in original equation
200/20 = 10
200/25 = 8
-----------
differ by 2 (thousand)

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let P = number of people in the club.
let C = Cost for each person
let T = total cost = 200000
-----
P*C = 200000 (equation 1)
-----
add 5 people to split the cost and save 2000 per person
-----
(P+5) * (C-2000) = 200000 (equation 2)
-----
you have 2 equations to work with:
-----
multiply out the second equation:
-----
P*C - 2000*P + 5*C - 10000 = 200000
since P*C equals 200000, then this equation also equals P*C
equation becomes:
P*C - 2000*P + 5*C - 10000 = P*C
subtract P*C from both sides of the equation:
-2000*P + 5*C - 10000 = 0
add 2000*P + 10000 to both sides of the equation:
5*C = 2000*P + 10000
divide both sides of the equation by 5:
C = 400*P + 2000
-----
in equation 1, substitute 400*P + 2000 for C
that equation becomes:
P * (400*P + 2000) = 200000
multiply out:
400*P^2 + 2000*P = 200000
divide both sides of equation by 400:
P^2 + 5*P = 500
subtract 500 from both sides of equation:
P^2 + 5*P - 500 = 0
solve the quadratic equation:
(P+25) * (P-20) = 0
roots are:
P = -25
or
P = 20
since the number of people can't be negative, P looks like it will be 20.
-----
if P = 20, then P*C = 200000 becomes 20*C = 200000 which make C = 10000.
likes like original number of people is 20 and original investment each is 10000 dollars.
-----
P = number of original members = 20.
C = original investment = 10000.
add 5 new members and total members = 25.
divide 200000 by 25 and you get 8000 investment each.
-----
number of members is 5 more than original (20 + 5 = 25).
amount of investment of each is 2000 less than original investment (10000 - 2000 = 8000).
-----

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domain = All numbers
range = All numbers

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f(x)=16–x^2
(graph 300x200 pixels, x from -6 to 5, y from -10 to 20, -x^2 +16).
Vertex y=16
y=(0,16).
y Intercept=16.
x intercept=+-4.

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sqrt 6 /( 4 sqrt 3 + sqrt 2)
=
=
=
=
.
Ed

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Find the exact length of the side of a square whose diagonal is 3 feet.
We are given the diagonal to be 3 feet but no side of the square is given.
Let x = side of the square.
Let 3 = the hypotenuse
We use the Pythagorean Theorem to find x.
x^2 + x^2 = 3^2
2x^2 = 9
Divide both sides by 2.
x^2 = 9/2
Take the square root of both sides.
x = sqrt{9/2}
x = sqrt{9}/sqrt{2}
x = 3/sqrt{2}
We rationalize the denominator.
Final answer:
x = 3(sqrt{2})/2

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restated so x=-4 and 3

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a^4 – 5a^2 + 4 = 0

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a(2) = - 16(2)^2 + 32(2) + 6 substitute "2" for "x" and solve
a(2)=6

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x^2 + 6x + 6 = 0
---------------

Solved by pluggable solver: SOLVE quadratic equation (work shown, graph etc)

Quadratic equation (in our case ) has the following solutons: For these solutions to exist, the discriminant should not be a negative number. First, we need to compute the discriminant : . Discriminant d=12 is greater than zero. That means that there are two solutions: . Quadratic expression can be factored: Again, the answer is: -1.26794919243112, -4.73205080756888. Here's your graph:

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If a manufacturer charges q dollars each for footballs, then he can sell 3000 - 150q footballs per week. Find a polynomial R(q) that represents the revenue for one week. Find the weekly revenue if the price is $8 for each football.
------------------
R(q) = 3000-150q
R(8) = 3000-150*8 = $1800.00
===============
Cheers,
Stan H.

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we will call the small side x and the larger side x+3. Now using the pathaorean formula we know that lets multiply these terms out combining terms and putting everything on the left we have using the quadratic formula we get x=9 and -12...we throw out the negative value and are left with x which is the amall side of 9, and the large side is x+3, which equals 12
small side = 9
large side = 12

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W= 5 (3sqrt