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 Polynomials-and-rational-expressions/133259: I need help trying to solve this. 6x-18/2x^2-5x+3* 4x^2+4x+1/ 6x+31 solutions Answer 97470 by jim_thompson5910(28595)   on 2008-03-22 21:30:44 (Show Source): You can put this solution on YOUR website! Start with the given expression Factor to get and factor 6 to get Factor to get Factor to get Factor to get Combine the fractions Cancel like terms Simplify -------------------------------------------------- Answer: So simplifies to . In other words
 Exponential-and-logarithmic-functions/133258: Use logs on both sides of the equation 9^28 - 9^27 / 8 = 3^x to find the unknown Please show all steps The 8 is dividing (9^28 - 9^27)1 solutions Answer 97468 by jim_thompson5910(28595)   on 2008-03-22 21:23:41 (Show Source): You can put this solution on YOUR website! Start with the given equation Rewrite into Break up into through the identity Factor out the GCF Evaluate to get 9 Subtract Cancel like terms Simplify Rewrite as Multiply the exponents. Remember, Multiply Since the bases are equal, the exponents are equal. So this means that
 Inequalities/133236: 12x + 5 ≤ 30 -2x Help me please??!!1 solutions Answer 97430 by jim_thompson5910(28595)   on 2008-03-22 19:02:11 (Show Source): You can put this solution on YOUR website! Start with the given inequality Subtract 5 from both sides Add 2x to both sides Combine like terms on the left side Combine like terms on the right side Divide both sides by 14 to isolate x -------------------------------------------------------------- Answer: So our answer is (which is approximately in decimal form)
 Radicals/133228: Radical Expressions simplify. I am trying to help my daughter with her homework and I am at a loss. I have never seen problems like these, and surely am at a loss as where to start with them. \/810+\/240-\/2501 solutions Answer 97427 by jim_thompson5910(28595)   on 2008-03-22 16:58:46 (Show Source): You can put this solution on YOUR website! Start with the given expression Simplify to get . Note: If you need help with simplifying square roots, check out this solver. Simplify to get . Simplify to get . Combine like terms ----------------------------------------- Answer: So simplifies to . In other words,
 Linear-equations/133226: Please I am in dire need of this answer. it's urgent please. Suppose a car rental company charges $150 for the first day and$100 for each additional or partial day. Let S(x) represent the cost of renting a car for x days. Find the value of S(5.5). Happy easter. 1 solutions Answer 97425 by jim_thompson5910(28595)   on 2008-03-22 16:06:12 (Show Source): You can put this solution on YOUR website!Since the person has to pay $150 no matter what, this is the fixed cost. Also, since the cost for each day is$100, this is the variable cost. So the equation would follow the form S(x)=Variable Cost * x + Fixed Cost So in this case the equation is Now plug in to evaluate S(5.5) Round 5.5 to 6 (remember, it's $100 per day or partial day) Multiply Add So this means that when x=5.5 the cost is$750
 Graphs/133213: Find the equation of the line through the given pair of points in standard form using only integers? (-4,-1) and (5,8)1 solutions Answer 97418 by jim_thompson5910(28595)   on 2008-03-22 14:57:13 (Show Source): You can put this solution on YOUR website!First lets find the slope through the points (,) and (,) Start with the slope formula (note: is the first point (,) and is the second point (,)) Plug in ,,, (these are the coordinates of given points) Subtract the terms in the numerator to get . Subtract the terms in the denominator to get Reduce So the slope is ------------------------------------------------ Now let's use the point-slope formula to find the equation of the line: ------Point-Slope Formula------ where is the slope, and is one of the given points So lets use the Point-Slope Formula to find the equation of the line Plug in , , and (these values are given) Rewrite as Rewrite as Distribute Multiply and to get Subtract from both sides to isolate y Combine like terms and to get ------------------------------------------------------------------------------------------------------------ Answer: So the equation of the line which goes through the points (,) and (,) is: The equation is now in form (which is slope-intercept form) where the slope is and the y-intercept is Notice if we graph the equation and plot the points (,) and (,), we get this: (note: if you need help with graphing, check out this solver) Graph of through the points (,) and (,) Notice how the two points lie on the line. This graphically verifies our answer.
Points-lines-and-rays/133221: Write the equation of the line perpendicular to y=2x-8 that passes through the point (4,0).
1 solutions

Answer 97417 by jim_thompson5910(28595)   on 2008-03-22 14:54:12 (Show Source):
You can put this solution on YOUR website!
 Solved by pluggable solver: Finding the Equation of a Line Parallel or Perpendicular to a Given Line Remember, any two perpendicular lines are negative reciprocals of each other. So if you're given the slope of , you can find the perpendicular slope by this formula: where is the perpendicular slope So plug in the given slope to find the perpendicular slope When you divide fractions, you multiply the first fraction (which is really ) by the reciprocal of the second Multiply the fractions. So the perpendicular slope is So now we know the slope of the unknown line is (its the negative reciprocal of from the line ). Also since the unknown line goes through (4,0), we can find the equation by plugging in this info into the point-slope formula Point-Slope Formula: where m is the slope and (,) is the given point Plug in , , and Distribute Multiply Add to both sides to isolate y Make into equivalent fractions with equal denominators Combine the fractions Reduce any fractions So the equation of the line that is perpendicular to and goes through (,) is So here are the graphs of the equations and graph of the given equation (red) and graph of the line (green) that is perpendicular to the given graph and goes through (,)

 Graphs/133211: Find the distance between the points, and find the midpoint of the line segment joining them? (9,8) and (2,1) 1 solutions Answer 97409 by jim_thompson5910(28595)   on 2008-03-22 14:13:43 (Show Source): You can put this solution on YOUR website!First let's find the distance Start with the given distance formula where is the first point and is the second point Plug in , , , Evaluate to get 7. Evaluate to get 7. Square each value Add Simplify the square root (note: If you need help with simplifying the square root, check out this solver) So the distance approximates to which rounds to 9.89949 So the distance between (9,8) and (2,1) is approximately 9.89949 units Now let's find the midpoint In order to find the midpoint between the points (9,8) and (2,1), we need to average each corresponding coordinate. In other words, we need to add up the corresponding coordinates and divide the sum by 2. So lets find the averages between the two points To find , average the x-coordinates between the two points So the x-coordinate of the midpoint is 5.5 (i.e. x=5.5) ----------------------------------------------------------------------------------------------------------------- To find , average the y-coordinates between the two points So the y-coordinate of the midpoint is 4.5 (i.e. y=4.5) ----------------------------------------------------------------------------------------------------------------- Answer: Since the coordinates of the midpoint are x=5.5, y=4.5, this means the midpoint is (5.5,4.5) Check: Here is a graph to visually see the answer Graph of the line segment with the endpoints (9,8) and (2,1) with the midpoint (5.5,4.5) We could visually verify our answer if we simply draw right triangles from each point like this: Here we can see that the two triangles are congruent (they both have a right angle and equal leg lengths), so our answer is verified.
 Linear-equations/133203: This question is from textbook algebra math 009 How do I resolve this equation: 7n+5=10n-101 solutions Answer 97402 by jim_thompson5910(28595)   on 2008-03-22 12:18:11 (Show Source): You can put this solution on YOUR website! Start with the given equation Subtract 5 from both sides Subtract 10n from both sides Combine like terms on the left side Combine like terms on the right side Divide both sides by -3 to isolate n Divide -------------------------------------------------------------- Answer: So our answer is
 Polynomials-and-rational-expressions/133100: Use rational exponents to simplify. root ( 3, x^6 ) 1 solutions Answer 97396 by jim_thompson5910(28595)   on 2008-03-22 11:43:16 (Show Source): You can put this solution on YOUR website! Start with the given expression Convert the expression from radical notation to exponent notation. Remember Now multiply the exponents Reduce So simplifies to . In other words, where x is positive.
 Polynomials-and-rational-expressions/133186: Use rational exponents to simplify. root (3,x^6)1 solutions Answer 97395 by jim_thompson5910(28595)   on 2008-03-22 11:42:34 (Show Source): You can put this solution on YOUR website! Start with the given expression Convert the expression from radical notation to exponent notation. Remember Now multiply the exponents Reduce So simplifies to . In other words, where x is positive.
 Quadratic_Equations/133182: Solve. x^2 + 6x -4 = 0 Here's what I think is the answer. Can someone tell me if I'm on the right track? (-6 + sqrt (52)/2) (-6 - sqrt ( 52 )/2)1 solutions Answer 97393 by jim_thompson5910(28595)   on 2008-03-22 11:37:26 (Show Source): You can put this solution on YOUR website!Let's use the quadratic formula to solve for x: Starting with the general quadratic the general solution using the quadratic equation is: So lets solve ( notice , , and ) Plug in a=1, b=6, and c=-4 Square 6 to get 36 Multiply to get Combine like terms in the radicand (everything under the square root) Simplify the square root (note: If you need help with simplifying the square root, check out this solver) Multiply 2 and 1 to get 2 So now the expression breaks down into two parts or Now break up the fraction or Simplify or ------------------------------------------- Answer: So the exact solutions are or So you are on the right track, but could simplify the answer.
 Coordinate-system/133167: x+5y=101 solutions Answer 97378 by jim_thompson5910(28595)   on 2008-03-22 00:19:53 (Show Source): You can put this solution on YOUR website!Do you want to solve for y? Start with the given equation Subtract from both sides Rearrange the equation Divide both sides by Break up the fraction Reduce So the equation is now in slope-intercept form () where the slope is and the y-intercept is
Polynomials-and-rational-expressions/133168: How do you factor the following problems step by step?
1. 9-x^2
2. 4x^2+81-36x
3. 4y^2+16y+16
4. x^-25/x+5
1 solutions

Answer 97376 by jim_thompson5910(28595)   on 2008-03-22 00:14:00 (Show Source):
You can put this solution on YOUR website!
I'll do the first two to get you started

# 1

Rewrite as

Rewrite as

Now use the difference of squares. Remember, the difference of squares formula is where in this case and

Plug in and

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

So the expression

factors to

Notice that if you foil the factored expression, you get the original expression. This verifies our answer.

Rearrange the terms

Looking at we can see that the first term is and the last term is where the coefficients are 4 and 81 respectively.

Now multiply the first coefficient 4 and the last coefficient 81 to get 324. Now what two numbers multiply to 324 and add to the middle coefficient -36? Let's list all of the factors of 324:

Factors of 324:
1,2,3,4,6,9,12,18,27,36,54,81,108,162

-1,-2,-3,-4,-6,-9,-12,-18,-27,-36,-54,-81,-108,-162 ...List the negative factors as well. This will allow us to find all possible combinations

These factors pair up and multiply to 324
1*324
2*162
3*108
4*81
6*54
9*36
12*27
18*18
(-1)*(-324)
(-2)*(-162)
(-3)*(-108)
(-4)*(-81)
(-6)*(-54)
(-9)*(-36)
(-12)*(-27)
(-18)*(-18)

note: remember two negative numbers multiplied together make a positive number

Now which of these pairs add to -36? Lets make a table of all of the pairs of factors we multiplied and see which two numbers add to -36

First NumberSecond NumberSum
13241+324=325
21622+162=164
31083+108=111
4814+81=85
6546+54=60
9369+36=45
122712+27=39
181818+18=36
-1-324-1+(-324)=-325
-2-162-2+(-162)=-164
-3-108-3+(-108)=-111
-4-81-4+(-81)=-85
-6-54-6+(-54)=-60
-9-36-9+(-36)=-45
-12-27-12+(-27)=-39
-18-18-18+(-18)=-36

From this list we can see that -18 and -18 add up to -36 and multiply to 324

Now looking at the expression , replace with (notice adds up to . So it is equivalent to )

Now let's factor by grouping:

Group like terms

Factor out the GCF of out of the first group. Factor out the GCF of out of the second group

Since we have a common term of , we can combine like terms

So factors to

So this also means that factors to (since is equivalent to )

note: is equivalent to since the term occurs twice. So also factors to

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So factors to

Linear-equations/133144: Find the equation, in standard form, with all integer coefficients, of the line perpendicular to x + 3y = 6 and passing through (-3, 5).

1 solutions

Answer 97340 by jim_thompson5910(28595)   on 2008-03-21 17:58:19 (Show Source):
You can put this solution on YOUR website!
First convert the standard equation into slope intercept form

 Solved by pluggable solver: Converting Linear Equations in Standard form to Slope-Intercept Form (and vice versa) Start with the given equation Subtract from both sides Multiply both sides by Distribute Multiply Rearrange the terms Reduce any fractions So the equation is now in slope-intercept form () where (the slope) and (the y-intercept)

Now let's find the equation of the line that is perpendicular to which goes through (-3,5)

 Solved by pluggable solver: Finding the Equation of a Line Parallel or Perpendicular to a Given Line Remember, any two perpendicular lines are negative reciprocals of each other. So if you're given the slope of , you can find the perpendicular slope by this formula: where is the perpendicular slope So plug in the given slope to find the perpendicular slope When you divide fractions, you multiply the first fraction (which is really ) by the reciprocal of the second Multiply the fractions. So the perpendicular slope is So now we know the slope of the unknown line is (its the negative reciprocal of from the line ). Also since the unknown line goes through (-3,5), we can find the equation by plugging in this info into the point-slope formula Point-Slope Formula: where m is the slope and (,) is the given point Plug in , , and Distribute Multiply Add to both sides to isolate y Combine like terms So the equation of the line that is perpendicular to and goes through (,) is So here are the graphs of the equations and graph of the given equation (red) and graph of the line (green) that is perpendicular to the given graph and goes through (,)

Now let's convert to standard form

Subtract 3x from both sides

Rearrange the terms

So the equation that is perpendicular to and goes through (-3,5) is

 Linear-equations/133143: Find the equation of the line that passes through the points (3, -2) and (4, -2). 1 solutions Answer 97339 by jim_thompson5910(28595)   on 2008-03-21 17:54:21 (Show Source): You can put this solution on YOUR website!First lets find the slope through the points (,) and (,) Start with the slope formula (note: is the first point (,) and is the second point (,)) Plug in ,,, (these are the coordinates of given points) Subtract the terms in the numerator to get . Subtract the terms in the denominator to get Reduce So the slope is ------------------------------------------------ Now let's use the point-slope formula to find the equation of the line: ------Point-Slope Formula------ where is the slope, and is one of the given points So lets use the Point-Slope Formula to find the equation of the line Plug in , , and (these values are given) Rewrite as Distribute Multiply and to get . Now reduce to get Subtract from both sides to isolate y Combine like terms and to get ------------------------------------------------------------------------------------------------------------ Answer: So the equation of the line which goes through the points (,) and (,) is: The equation is now in form (which is slope-intercept form) where the slope is and the y-intercept is Notice if we graph the equation and plot the points (,) and (,), we get this: (note: if you need help with graphing, check out this solver) Graph of through the points (,) and (,) Notice how the two points lie on the line. This graphically verifies our answer.
 Linear-equations/133142: Find the slope of the line that passes through the points (2, 3) and (5, 8). 1 solutions Answer 97338 by jim_thompson5910(28595)   on 2008-03-21 17:50:27 (Show Source): You can put this solution on YOUR website! Let's denote the first point (2,3) as . In other words, and Now let's denote the second point (5,8) as . In other words, and ------------------------- Start with the slope formula Plug in ,,, Subtract the terms in the numerator to get . Subtract the terms in the denominator to get --------------------- Answer: So the slope of the line through the points (2,3) and (5,8) is
 Probability-and-statistics/133138: Find the value of the logarithm without using a calculator In(e^5) Thank you1 solutions Answer 97337 by jim_thompson5910(28595)   on 2008-03-21 17:44:22 (Show Source): You can put this solution on YOUR website! Start with the given expression Rewrite the expression using the identity Replace with 1. Remember, Multiply So
 Square-cubic-other-roots/133120: Please help me with this problem. √(98x^3)1 solutions Answer 97330 by jim_thompson5910(28595)   on 2008-03-21 16:15:45 (Show Source): You can put this solution on YOUR website! Start with the given expression Break up the root Simplify to get Simplify to get Combine the roots ---------------------------------- Answer: So simplifies to In other words, where every variable is positive
 Polynomials-and-rational-expressions/133112: Find the following. Assume that variables can represent any real numbers. sqrt ( a+4 )^2 The a+4 should be in parenthesis with an exponent of 2 in a radical sign. I hope I explained that correctly.1 solutions Answer 97328 by jim_thompson5910(28595)   on 2008-03-21 15:59:10 (Show Source): You can put this solution on YOUR website!If you have something like , how do you solve this? Well you simply take the square root of both sides like this Notice how simplifies to . What's happening is the square root is undoing the square exponent. --------------------- So the same thing is happening with . If you can't see it, then let to get so this simplifies to where x is positive Now plug into x to get So this means that simplifies to . In other words, where
 real-numbers/133116: solve for x: 5(x-2)=2(10+x)1 solutions Answer 97323 by jim_thompson5910(28595)   on 2008-03-21 15:51:28 (Show Source): You can put this solution on YOUR website! Start with the given equation Distribute Add 10 to both sides Subtract 2x from both sides Combine like terms on the left side Combine like terms on the right side Divide both sides by 3 to isolate x Divide -------------------------------------------------------------- Answer: So our answer is
 Sequences-and-series/133104: This question is from textbook Use inductive reasoning to predict the next TWO numbers in the sequence. 80, –40, 20, –10, 5, . . . I can't figure out what the pattern is her necessarily. i get that its -120 +60 -30 +15 but i don't get what comes next. Help me out with this please.1 solutions Answer 97320 by jim_thompson5910(28595)   on 2008-03-21 15:34:52 (Show Source): You can put this solution on YOUR website!Notice how each term is cut in half and is negated (ie the sign is changed). In other words, each term is multiplied by Cut the 1st term 80 in half and negate it to get –40. In other words, Cut the 2nd term –40 in half and negate it to get 20. In other words, Cut the 3rd term 20 in half and negate it to get –10. In other words, Cut the 4th term –10 in half and negate it to get 5. In other words, So to find the next two terms, simply multiply the last few terms by So lets multiply the last term 5 by So the next term is or -2.5 Now lets multiply the last term by So the next term is or 1.25 So the updated sequence is 80, –40, 20, –10, 5, ,
Polynomials-and-rational-expressions/133108: Factor completely.
252r^2 + 588rf + 343f^2
Thank you for any help you can give me in figuring this problem out.
1 solutions

Answer 97318 by jim_thompson5910(28595)   on 2008-03-21 15:17:25 (Show Source):
You can put this solution on YOUR website!

Factor out the GCF

Now let's focus on the inner expression

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

Looking at we can see that the first term is and the last term is where the coefficients are 36 and 49 respectively.

Now multiply the first coefficient 36 and the last coefficient 49 to get 1764. Now what two numbers multiply to 1764 and add to the middle coefficient 84? Let's list all of the factors of 1764:

Factors of 1764:
1,2,3,4,6,7,9,12,14,18,21,28,36,42,49,63,84,98,126,147,196,252,294,441,588,882

-1,-2,-3,-4,-6,-7,-9,-12,-14,-18,-21,-28,-36,-42,-49,-63,-84,-98,-126,-147,-196,-252,-294,-441,-588,-882 ...List the negative factors as well. This will allow us to find all possible combinations

These factors pair up and multiply to 1764
1*1764
2*882
3*588
4*441
6*294
7*252
9*196
12*147
14*126
18*98
21*84
28*63
36*49
42*42
(-1)*(-1764)
(-2)*(-882)
(-3)*(-588)
(-4)*(-441)
(-6)*(-294)
(-7)*(-252)
(-9)*(-196)
(-12)*(-147)
(-14)*(-126)
(-18)*(-98)
(-21)*(-84)
(-28)*(-63)
(-36)*(-49)
(-42)*(-42)

note: remember two negative numbers multiplied together make a positive number

Now which of these pairs add to 84? Lets make a table of all of the pairs of factors we multiplied and see which two numbers add to 84

First NumberSecond NumberSum
117641+1764=1765
28822+882=884
35883+588=591
44414+441=445
62946+294=300
72527+252=259
91969+196=205
1214712+147=159
1412614+126=140
189818+98=116
218421+84=105
286328+63=91
364936+49=85
424242+42=84
-1-1764-1+(-1764)=-1765
-2-882-2+(-882)=-884
-3-588-3+(-588)=-591
-4-441-4+(-441)=-445
-6-294-6+(-294)=-300
-7-252-7+(-252)=-259
-9-196-9+(-196)=-205
-12-147-12+(-147)=-159
-14-126-14+(-126)=-140
-18-98-18+(-98)=-116
-21-84-21+(-84)=-105
-28-63-28+(-63)=-91
-36-49-36+(-49)=-85
-42-42-42+(-42)=-84

From this list we can see that 42 and 42 add up to 84 and multiply to 1764

Now looking at the expression , replace with (notice adds up to . So it is equivalent to )

Now let's factor by grouping:

Group like terms

Factor out the GCF of out of the first group. Factor out the GCF of out of the second group

Since we have a common term of , we can combine like terms

So factors to

So this also means that factors to (since is equivalent to )

note: is equivalent to since the term occurs twice. So also factors to

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

So our expression goes from and factors further to

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