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Quadratic_Equations/694634: 9x^2-36x+36 factored
1 solutions

Answer 428098 by jim_thompson5910(28595)   on 2012-12-15 20:06:34 (Show Source):
You can put this solution on YOUR website!

Start with the given expression.

Factor out the GCF .

Now let's try to factor the inner expression

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

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 :
1,2,4
-1,-2,-4

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

These factors pair up and multiply to .
1*4 = 4
2*2 = 4
(-1)*(-4) = 4
(-2)*(-2) = 4

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

First NumberSecond NumberSum
141+4=5
222+2=4
-1-4-1+(-4)=-5
-2-2-2+(-2)=-4

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

Condense the terms.

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

So then factors further to

===============================================================

Answer:

So completely factors to .

In other words, .

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

Expressions-with-variables/694628: Factor this trinomial:
2t^2 - t - 3
1 solutions

Answer 428087 by jim_thompson5910(28595)   on 2012-12-15 19:10:01 (Show Source):
You can put this solution on YOUR website!

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 :
1,2,3,6
-1,-2,-3,-6

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

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

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

First NumberSecond NumberSum
1-61+(-6)=-5
2-32+(-3)=-1
-16-1+6=5
-23-2+3=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 .

In other words, .

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

 Coordinate-system/694625: -3x-4y>16 into slope intercept form 1 solutions Answer 428076 by jim_thompson5910(28595)   on 2012-12-15 18:39:26 (Show Source): You can put this solution on YOUR website! Answer:
 Graphs/694623: I need someone to show me how to graph this inequality. Please help! x + y < -1 1 solutions Answer 428074 by jim_thompson5910(28595)   on 2012-12-15 18:38:16 (Show Source): You can put this solution on YOUR website!solve for y x + y < -1 y < -x-1 Now graph the line y = -x - 1. You do this by plotting the points (0, -1) and (1, -2) and drawing a straight line through them. Then make this straight line a dashed line. Then shade below this dashed line.
 Finance/694620: mr brand invested $7000, part at 8% and the rest at5% His total annual income from these investment was$500 find the amount invested at each rate 1 solutions Answer 428068 by jim_thompson5910(28595)   on 2012-12-15 18:14:47 (Show Source): You can put this solution on YOUR website!Hint: x+y = 7000 y = 7000 - x --------------------------- 0.08x + 0.05y = 500 8x + 5y = 50000 8x + 5(7000 - x) = 50000 Solve for x and then use this to solve for y.
 test/694614: x+y=-7 X-y=11 2x+y=6 X-y=-91 solutions Answer 428064 by jim_thompson5910(28595)   on 2012-12-15 18:07:30 (Show Source): You can put this solution on YOUR website!Are these equations all part of the same system? Or are they paired? Please elaborate. Thanks.
 Coordinate-system/694618: -2x-y<9 into slope intercept form1 solutions Answer 428062 by jim_thompson5910(28595)   on 2012-12-15 18:06:04 (Show Source): You can put this solution on YOUR website!-2x-y < 9 -y < 2x + 9 y > -2x - 9 Answer: y > -2x - 9
 logarithm/694594: write the expression as a single logarithm. 3log5(2)+log5(7) please show steps too thanks1 solutions Answer 428059 by jim_thompson5910(28595)   on 2012-12-15 18:02:09 (Show Source): You can put this solution on YOUR website!3log5(2)+log5(7) log5(2^3)+log5(7) log5(8)+log5(7) log5(8*7) log5(56) Answer: log5(56)
 test/694600: Log6 36=21 solutions Answer 428045 by jim_thompson5910(28595)   on 2012-12-15 16:30:37 (Show Source): You can put this solution on YOUR website!Log6 36=2 converts to
 Finite-and-infinite-sets/694596: what is the value for y when 5y^2-12y+35=0? 1 solutions Answer 428042 by jim_thompson5910(28595)   on 2012-12-15 16:13:08 (Show Source): You can put this solution on YOUR website! Plug in , , or or or
 Points-lines-and-rays/694581: x+5 3x-1, 4x-2, perimeter=66 find the length of the shortest side1 solutions Answer 428032 by jim_thompson5910(28595)   on 2012-12-15 14:57:45 (Show Source): You can put this solution on YOUR website!Hint: Add up the three sides and set that sum equal to the perimeter (Side1) + (Side2) + (Side3) = Perimeter (x+5) + (3x-1) + (4x-2) = 66 x+5 + 3x-1 + 4x-2 = 66 8x + 2 = 66 I'll let you finish
 Coordinate-system/694583: What is an equation in slope intercept form when the coordinates are (-2, -3) and zero slope?1 solutions Answer 428031 by jim_thompson5910(28595)   on 2012-12-15 14:55:04 (Show Source): You can put this solution on YOUR website!y = mx + b y = 0x + b y = b y = -3 Answer: y = -3
 Quadratic_Equations/694585: is this true or false: 4^x*0=01 solutions Answer 428030 by jim_thompson5910(28595)   on 2012-12-15 14:54:25 (Show Source): You can put this solution on YOUR website!It's true because any number times zero is zero.
 Roman-numerals/694584: XXIV in numbers1 solutions Answer 428029 by jim_thompson5910(28595)   on 2012-12-15 14:53:13 (Show Source): You can put this solution on YOUR website!XXIV = 10+10+4 XXIV = 24
 Permutations/694415: 1 solutions Answer 427905 by jim_thompson5910(28595)   on 2012-12-14 18:23:55 (Show Source): You can put this solution on YOUR website!problem missing
 Polygons/694431: LM is the midsegment of trapezoid ABCD. If AB = 46 and DC = 125, what is LM? 1 solutions Answer 427904 by jim_thompson5910(28595)   on 2012-12-14 18:23:23 (Show Source): You can put this solution on YOUR website!Average the two to get (46+125)/2 = 85.5
 Linear-equations/694421: Find an equation of the line containing the given points (-7,-8)and (-4,-6)1 solutions Answer 427886 by jim_thompson5910(28595)   on 2012-12-14 17:21:21 (Show Source): You can put this solution on YOUR website! First let's find the slope of the line through the points and Note: is the first point . So this means that and . Also, is the second point . So this means that and . Start with the slope formula. Plug in , , , and Subtract from to get Subtract from to get So the slope of the line that goes through the points and is Now let's use the point slope formula: Start with the point slope formula Plug in , , and Rewrite as Rewrite as Distribute Multiply Subtract 8 from both sides. Combine like terms. note: If you need help with fractions, check out this solver. So the equation that goes through the points and is
 Linear-equations/694422: Write an equation of a line with slope of -3 and y-intercept (0,-2). Give it in the form Ax+By=C.1 solutions Answer 427884 by jim_thompson5910(28595)   on 2012-12-14 17:13:28 (Show Source): You can put this solution on YOUR website!y = mx + b y = -3x - 2 3x + y = -2 Answer: 3x + y = -2
 Rational-functions/694383: Use the Binomial Theorem to expand the binomial. (3v + s)51 solutions Answer 427881 by jim_thompson5910(28595)   on 2012-12-14 17:05:48 (Show Source): You can put this solution on YOUR website!Look at Row 6 in Pascals triangle to see these numbers: 1, 5, 10, 10, 5, 1 These coefficients multiply with the following terms: (3v)^5*(s)^0 (3v)^4*(s)^1 (3v)^3*(s)^2 (3v)^2*(s)^3 (3v)^1*(s)^4 (3v)^0*(s)^5 So multiply them to get the following 1*(3v)^5*(s)^0 5*(3v)^4*(s)^1 10*(3v)^3*(s)^2 10*(3v)^2*(s)^3 5*(3v)^1*(s)^4 1*(3v)^0*(s)^5 1*(243v^5)*(1) 5*(81v^4)*(s) 10*(27v^3)*(s^2) 10*(9v^2)*(s^3) 5*(3v)*(s^4) 1*(1)*(s^5) 243v^5 405sv^4 270s^2v^3 90s^3v^2 15s^4v s^5 Then add up all the terms to get this final answer 243v^5 + 405sv^4 + 270s^2v^3 + 90s^3v^2 + 15s^4v + s^5 So (3v+s)^5 = 243v^5 + 405sv^4 + 270s^2v^3 + 90s^3v^2 + 15s^4v + s^5
 test/694416: 1 solutions Answer 427880 by jim_thompson5910(28595)   on 2012-12-14 17:02:49 (Show Source): You can put this solution on YOUR website!question missing, please repost. Thanks.
 Polynomials-and-rational-expressions/694419: Eight inches of heavy, wet snow are equivalent to 4 inches of rain. Estimate the water content of 12 inches of heavy, wet snow.1 solutions Answer 427879 by jim_thompson5910(28595)   on 2012-12-14 17:02:30 (Show Source): You can put this solution on YOUR website!8/4 = 12/x 8x = 4*12 8x = 48 x = 48/8 x = 6 so the answer is 6
 Rate-of-work-word-problems/694392: If a painter can paint a house in 6 hours and his apprentice can paint the same sized house in 9 hours, how long will it take for them to paint a house if they worked together?" 1 solutions Answer 427851 by jim_thompson5910(28595)   on 2012-12-14 15:18:50 (Show Source): You can put this solution on YOUR website!1/6 + 1/9 = 1/t 3/18 + 2/18 = 1/t 5/18 = 1/t 5t = 18 t = 18/5 So it will take them 18/5 hours, or 3 and 3/5 hours (or 3.6 hours = 3 hours and 36 minutes) if they work together.
 Equations/694347: Need help with my math this one -(s-5)1 solutions Answer 427845 by jim_thompson5910(28595)   on 2012-12-14 14:08:29 (Show Source): You can put this solution on YOUR website!-(s-5) distributes to -s + 5
 test/694366: (8-6i) + (7 + 4i)1 solutions Answer 427844 by jim_thompson5910(28595)   on 2012-12-14 14:07:55 (Show Source): You can put this solution on YOUR website!(8-6i) + (7 + 4i) 8-6i + 7 + 4i 15 - 2i
 Divisibility_and_Prime_Numbers/694189: Factor x^2 - 91 solutions Answer 427787 by jim_thompson5910(28595)   on 2012-12-13 21:58:01 (Show Source): You can put this solution on YOUR website!x^2 - 9 = (x + 3)(x - 3)
 Parallelograms/694176: If you can not answer all of my question it is fine but i would like some help on these problems: The choices for the answer are True or False: 1.If three points are collinear, then they are coplanar 2.Any two points are collinear 3.Given two points, there is more than one plane containing them 4. The bisector of the vertex angle of an isosceles triangle bisects the base and is perependicular to the base 5. The base angles of an isosceles triangle are acute 6. An altitude of a triangle lies in the interior of the triangle 7. If three triangles have unequal measures, then no two sides of the triangle are congruent 8. Every right angle has two acute angles 9. The longest side of any triangle is called the hypotenuse 10. The median of a triangle bisects the side to which its drawn 11. Each side of an angle is a ray 12. The diagonals of a square are perpendicular to each other. Thanks1 solutions Answer 427778 by jim_thompson5910(28595)   on 2012-12-13 21:36:46 (Show Source): You can put this solution on YOUR website!I'll do the first six to get you started. 1.If three points are collinear, then they are coplanar ... false (think of a line going through a plane but the line isn't all in the plane) 2.Any two points are collinear ... true by definition 3.Given two points, there is more than one plane containing them ... true, you need 3 points to uniquely determine a plane 4. The bisector of the vertex angle of an isosceles triangle bisects the base and is perependicular to the base ... true, this takes a while to explain and prove, but it can be done. 5. The base angles of an isosceles triangle are acute ... true, a picture may help see this 6. An altitude of a triangle lies in the interior of the triangle ... false, this is sometimes the case, but sometimes not. Think obtuse triangles. I'll let you do the rest. 7. If three triangles have unequal measures, then no two sides of the triangle are congruent 8. Every right angle has two acute angles 9. The longest side of any triangle is called the hypotenuse 10. The median of a triangle bisects the side to which its drawn 11. Each side of an angle is a ray 12. The diagonals of a square are perpendicular to each other.
 Complex_Numbers/694174: find all 12 solutions to x^12=1 Already know 1 and -1 are solutions, but I'm not sure how to factor the equation.1 solutions Answer 427775 by jim_thompson5910(28595)   on 2012-12-13 21:32:51 (Show Source): You can put this solution on YOUR website!Hint: x^12=1 x^12 - 1 = 0 (x^4)^3 - 1^3 = 0 (x^4 - 1)((x^4)^2 + x^4*1 + 1^2) = 0 (x^4 - 1)(x^8 + x^4 + 1) = 0 (x^2 - 1)(x^2 + 1)(x^8 + x^4 + 1) = 0 (x - 1)(x + 1)(x^2 + 1)(x^8 + x^4 + 1) = 0 Note: To solve x^8 + x^4 + 1 = 0, let z = x^4 to get z^2 + z + 1 = 0. Solve for z and use this to find solutions to x.
 Inequalities/694175: -5p-4+6p>12 i need help1 solutions Answer 427773 by jim_thompson5910(28595)   on 2012-12-13 21:27:31 (Show Source): You can put this solution on YOUR website! Start with the given inequality. Combine like terms on the left side. Add to both sides. Combine like terms on the right side. ---------------------------------------------------------------------- Answer: So the solution is
 Angles/694165: If you can not answer all of my question it is fine but i would like some help on these problems: The choices for the answer are Sometimes, Always, or Never: 1. A ray has two endpoints 2. An angle is equal to its supplement 3. A right triangle is isosceles 4. The complement of an acute angle is an obtuse angle 5. Two lines perpendicular to a third line are skew 6. Two skew lines are coplanar 7. If two angles have a common side, the angles are adjacent 8. The supplement of an obtuse angle is an obtuse angle Thanks1 solutions Answer 427767 by jim_thompson5910(28595)   on 2012-12-13 21:08:55 (Show Source): You can put this solution on YOUR website!1. A ray has two endpoints .... never It only has one endpoint. Always. ------------------------------------------------------- 2. An angle is equal to its supplement Sometimes and this is only true if you have a right angle ------------------------------------------------------- 3. A right triangle is isosceles sometimes only if you have a right isosceles triangle ------------------------------------------------------- 4. The complement of an acute angle is an obtuse angle This is never true draw a picture to see this, but you'll see the other angle isn't obtuse since it's acute as well. ------------------------------------------------------- 5. Two lines perpendicular to a third line are skew This is sometimes true. It depends on if you are in 3d space or the plane. If you are in the plane, then it's never true. However I'm saying "sometimes" because I'm thinking the problem is thinking 3d space. ------------------------------------------------------- 6. Two skew lines are coplanar This is never true because if it were, then the two lines would intersect and not be skew. ------------------------------------------------------- I'll let you tackle the last two.
 Functions/694143: if f(x)= 4x-x^2 and g(x)= 1/x, then (fog)(1/2) is equal to 1 solutions Answer 427763 by jim_thompson5910(28595)   on 2012-12-13 20:54:14 (Show Source): You can put this solution on YOUR website!g(x) = 1/x g(1/2) = 1/(1/2) g(1/2) = (1/1)*(2/1) g(1/2) = 2 ------------------------------------------------------- f(x) = 4x - x^2 f(g(x)) = 4(g(x)) - (g(x))^2 f(g(1/2)) = 4(g(1/2)) - (g(1/2))^2 f(g(1/2)) = 4(2) - (2)^2 f(g(1/2)) = 4(2) - 4 f(g(1/2)) = 8 - 4 f(g(1/2)) = 4 (f o g)(1/2) = 4
 Functions/694160: Okay, so in general g(x)=x^2-2x, and the problem is 2(g(t))1 solutions Answer 427761 by jim_thompson5910(28595)   on 2012-12-13 20:52:22 (Show Source): You can put this solution on YOUR website!g(x)=x^2-2x g(t)=t^2-2t 2*g(t)=2*(t^2-2t) 2*g(t) = 2t^2-4t