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Radicals/195955: Rationalize the denominator 2/(square root[6] - square root [5]) If you could help me with this, i would appreciate it. 1 solutions Answer 146933 by jim_thompson5910(28593)   on 2009-05-11 23:11:06 (Show Source): You can put this solution on YOUR website! Start with the given expression. Multiply both the numerator and denominator by FOIL the denominator (use the difference of squares formula) Square each term. Combine like terms. Reduce Distribute So

Rational-functions/195959: Divide: [(2x^2 + 5x -12)/(9x^2 - 16)]/[(2x^2 - 7x + 6)/3x^2 - x - 4)] Any help will be appreciated. Thanks 1 solutions Answer 146931 by jim_thompson5910(28593)   on 2009-05-11 23:08:13 (Show Source): You can put this solution on YOUR website! Start with the given expression. Multiply the first fraction by the reciprocal of the second fraction . Factor to get . Factor to get . Factor to get . Factor to get . Combine the fractions. Highlight the common terms. Cancel out the common terms. Simplify. FOIL So simplifies to . In other words, where , , , or

Numbers_Word_Problems/195930: The sum of the squares of three consecutive, positive integers is equal to the sum of the squares of the next two integers. Find the five integers. 1 solutions Answer 146921 by jim_thompson5910(28593)   on 2009-05-11 21:48:06 (Show Source): You can put this solution on YOUR website! Consecutive integers follow the form: x, x+1, x+2, x+3, etc... So... "The sum of the squares of three consecutive, positive integers is equal to the sum of the squares of the next two integers." translates to Start with the given equation. FOIL Combine like terms. Get all terms to the left side. Combine like terms. Notice we have a quadratic equation in the form of where , , and Let's use the quadratic formula to solve for x Start with the quadratic formula Plug in , , and Negate to get . Square to get . Multiply to get Rewrite as Add to to get Multiply and to get . Take the square root of to get . or Break up the expression. or Combine like terms. or Simplify. So the answers are or Since the problem mentions that the numbers are positive. So this means that the only solution is This means that the numbers are: 10, 11, 12, 13, and 14

Problems-with-consecutive-odd-even-integers/195949: The sum of the squares of three consecutive, positive integers is equal to the sum of the squares of the next two integers. Find the five integers. x + x +2 + x + 3 = x squared + x squared.. is the best I can come up with???? help! Thanks. 1 solutions Answer 146919 by jim_thompson5910(28593)   on 2009-05-11 21:46:08 (Show Source): You can put this solution on YOUR website! Consecutive integers follow the form: x, x+1, x+2, x+3, etc... So... "The sum of the squares of three consecutive, positive integers is equal to the sum of the squares of the next two integers." translates to Start with the given equation. FOIL Combine like terms. Get all terms to the left side. Combine like terms. Notice we have a quadratic equation in the form of where , , and Let's use the quadratic formula to solve for x Start with the quadratic formula Plug in , , and Negate to get . Square to get . Multiply to get Rewrite as Add to to get Multiply and to get . Take the square root of to get . or Break up the expression. or Combine like terms. or Simplify. So the answers are or Once again, the problem mentions that the numbers are positive. So this means that the only solution is This means that the numbers are: 10, 11, 12, 13, and 14

Graphs/195950: Solve: (20b^3+17b^2+18b+42)/(4b+5) 1 solutions Answer 146917 by jim_thompson5910(28593)   on 2009-05-11 21:39:06 (Show Source): You can put this solution on YOUR website! Simply use polynomial long division: Since the quotient is and the remainder is 7, this means that

Radicals/195946: Please help me simplify this radical. 3 Square root 6 minus 3 square root 24 1 solutions Answer 146915 by jim_thompson5910(28593)   on 2009-05-11 21:21:56 (Show Source): You can put this solution on YOUR website! Start with the given expression Simplify to get . Multiply 3 and 2 to get 6. Combine like terms. So simplifies to . In other words,

Linear-systems/195945: Find all the solutions of the following systems of equations if possible. If a system has no solution, explain why it does not. (a) 3X – Y = 3 (b) 2X – Y = 1 (c) 2X – Y = 1 3X +Y = 15 2Y – 4X = 3 2Y – 4X = (-2) 1 solutions Answer 146914 by jim_thompson5910(28593)   on 2009-05-11 21:19:47 (Show Source): You can put this solution on YOUR website! I'll do the two problems to get you started A) Start with the given system of equations: Add the equations together. You can do this by simply adding the two left sides and the two right sides separately like this: Group like terms. Combine like terms. Simplify. Divide both sides by to isolate . Reduce. ------------------------------------------------------------------ Now go back to the first equation. Plug in . Multiply. Subtract from both sides. Combine like terms on the right side. Divide both sides by to isolate . Reduce. So the solutions are and . Which form the ordered pair . This means that the system is consistent and independent. Notice when we graph the equations, we see that they intersect at . So this visually verifies our answer. Graph of (red) and (green) B) Start with the second equation. Rearrange the terms. So we have the system of equations: Multiply the both sides of the first equation by 2. Distribute and multiply. So we have the new system of equations: Now add the equations together. You can do this by simply adding the two left sides and the two right sides separately like this: Group like terms. Combine like terms. Simplify. Since is never true, this means that there are no solutions. So the system is inconsistent.

Radicals/195938: 2(sqrt of 3 + sqrt of 12) 1 solutions Answer 146912 by jim_thompson5910(28593)   on 2009-05-11 21:13:01 (Show Source): You can put this solution on YOUR website! Start with the given expression. Distribute Factor Break up the square root Take the square root of 4 to get 2 Multiply Combine like terms. So

Polynomials-and-rational-expressions/195928: Find P(-1/2) if P(x)= 4x^4 - 2x^3 + 17 1 solutions Answer 146911 by jim_thompson5910(28593)   on 2009-05-11 21:10:35 (Show Source): You can put this solution on YOUR website! Start with the given equation. Plug in . Raise to the 4th power to get . Cube to get . Multiply and to get . Multiply and to get . Combine like terms. So the answer is

Numbers_Word_Problems/195933: Twice the square of a certain postive number is 144 more than twice the number. What is the number? Really looking for help on how to set up.... can solve but can't seem to figure out how to set the thing up. Thanks. 1 solutions Answer 146910 by jim_thompson5910(28593)   on 2009-05-11 21:08:35 (Show Source): You can put this solution on YOUR website! You're probably hung up on the translation: "Twice the square of a certain postive number" ----> "is" ---> = "144 more than twice the number" ----> So the full translation is Start with the given equation. Get all terms to the left side. Notice we have a quadratic equation in the form of where , , and Let's use the quadratic formula to solve for x Start with the quadratic formula Plug in , , and Negate to get . Square to get . Multiply to get Rewrite as Add to to get Multiply and to get . Take the square root of to get . or Break up the expression. or Combine like terms. or Simplify. Remember, the problem stated that the number is positive. So the only answer is

Radicals/195939: sqrt of 3(2+sqrt of 12) 1 solutions Answer 146909 by jim_thompson5910(28593)   on 2009-05-11 21:04:49 (Show Source): You can put this solution on YOUR website! Start with the given expression. Distribute Rearrange the terms. Combine the roots. Multiply Take the square root of 36 to get 6 Rearrange the terms. So

Numbers_Word_Problems/195935: The square of a positive number decreased by 10 is 2 more than 4 times the number. What is the number? Trouble setting up.... any helpful sites that work with this type of question or one that can teach me? . Thanks.... frustrated mom:) 1 solutions Answer 146908 by jim_thompson5910(28593)   on 2009-05-11 21:01:32 (Show Source): You can put this solution on YOUR website! "The square of a positive number decreased by 10 is 2 more than 4 times the number" translates to Start with the given equation. Get all terms to the left side. Combine like terms. Notice we have a quadratic equation in the form of where , , and Let's use the quadratic formula to solve for x Start with the quadratic formula Plug in , , and Negate to get . Square to get . Multiply to get Rewrite as Add to to get Multiply and to get . Take the square root of to get . or Break up the expression. or Combine like terms. or Simplify. Since it is specified that the number is positive, this means that the only answer is

Percentage-and-ratio-word-problems/195885: Evaluate 3/4+1/2 1 solutions Answer 146874 by jim_thompson5910(28593)   on 2009-05-11 17:17:03 (Show Source): You can put this solution on YOUR website! So

Expressions-with-variables/195871: Use the quadratic formula to solve 2y2 - 6y - 8 = 0. 1 solutions Answer 146872 by jim_thompson5910(28593)   on 2009-05-11 17:16:09 (Show Source): You can put this solution on YOUR website! Start with the given equation. Notice we have a quadratic equation in the form of where , , and Let's use the quadratic formula to solve for y Start with the quadratic formula Plug in , , and Negate to get . Square to get . Multiply to get Rewrite as Add to to get Multiply and to get . Take the square root of to get . or Break up the expression. or Combine like terms. or Simplify. So the answers are or

Polynomials-and-rational-expressions/195881: Solve by the method of your choice x^2-6x-7=0 1 solutions Answer 146870 by jim_thompson5910(28593)   on 2009-05-11 17:14:05 (Show Source): You can put this solution on YOUR website! Start with the given equation. Notice we have a quadratic equation in the form of where , , and Let's use the quadratic formula to solve for x Start with the quadratic formula Plug in , , and Negate to get . Square to get . Multiply to get Rewrite as Add to to get Multiply and to get . Take the square root of to get . or Break up the expression. or Combine like terms. or Simplify. So the answers are or

Square-cubic-other-roots/195883: Solve the proportion -3/4=5/x 1 solutions Answer 146869 by jim_thompson5910(28593)   on 2009-05-11 17:09:02 (Show Source): You can put this solution on YOUR website! Start with the given ratio Multiply both sides by x Multiply both sides by 4 Multiply Divide both sides by -3 Reduce. So our answer is which is roughly

Polynomials-and-rational-expressions/195843: Multiply and Simplify: 2x^2-x-3 over x^2-1 * x^2+x-2 over 2x^2 +x-6 1 solutions Answer 146857 by jim_thompson5910(28593)   on 2009-05-11 14:03:39 (Show Source): You can put this solution on YOUR website! Start with the given expression. Factor to get . Factor to get . Factor to get . Factor to get . Combine the fractions. Highlight the common terms. Cancel out the common terms. Simplify. So simplifies to . In other words, where , , , or

logarithm/195840: I have a question that deals with Logarithmic Functions. I have a work sheet and the directions say to "solve". the problem is 2 with and exponet of "x" =81. 1 solutions Answer 146852 by jim_thompson5910(28593)   on 2009-05-11 13:42:58 (Show Source): You can put this solution on YOUR website! Start with the given equation Take the log of both sides. Pull down the exponent. Divide both sides by Use the change of base formula So the solution is which approximates to

Radicals/195791: Divide: 2x2 + 5x – 12 ÷ 2x2 – 7x + 6 9x2 – 16 3x2 – x – 4 Should be 2x^2+5x-12 over 9x^2-16/ 2x^2-7x+6 over 3x^2-x-4 1 solutions Answer 146843 by jim_thompson5910(28593)   on 2009-05-11 13:13:43 (Show Source): You can put this solution on YOUR website! Start with the given expression. Multiply the first fraction by the reciprocal of the second fraction . Factor to get . Factor to get . Factor to get . Factor to get . Combine the fractions. Highlight the common terms. Cancel out the common terms. Simplify. FOIL So simplifies to . In other words, where , , , , or

Polynomials-and-rational-expressions/195821: What are the factors of n^2-7n+10 1 solutions Answer 146840 by jim_thompson5910(28593)   on 2009-05-11 13:09:13 (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 : 1,2,5,10 -1,-2,-5,-10 Note: list the negative of each factor. This will allow us to find all possible combinations. These factors pair up and multiply to . 1*10 2*5 (-1)*(-10) (-2)*(-5) 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 10 1+10=11 2 5 2+5=7 -1 -10 -1+(-10)=-11 -2 -5 -2+(-5)=-7 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).

Polynomials-and-rational-expressions/195822: WHat are the factors of x^2-8x+12 1 solutions Answer 146839 by jim_thompson5910(28593)   on 2009-05-11 13:08:37 (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 : 1,2,3,4,6,12 -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=13 2 6 2+6=8 3 4 3+4=7 -1 -12 -1+(-12)=-13 -2 -6 -2+(-6)=-8 -3 -4 -3+(-4)=-7 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).

Polynomials-and-rational-expressions/195828: Please help! I need to express the following rational expression in lowest terms: x^3+5x^2 --------- 2x^2+9x-5 1 solutions Answer 146837 by jim_thompson5910(28593)   on 2009-05-11 13:07:39 (Show Source): You can put this solution on YOUR website! The key to these types of problems is factoring and canceling out common terms. Start with the given expression. Factor to get . Factor to get . Highlight the common terms. Cancel out the common terms. Simplify. So simplifies to . In other words, where or

Equations/195802: I need to divide and simplify this problem. x^2-1 4x-4 ------ / ----- x^2-14x+49 x^2-2x-35 Thank you for your help!!!!! 1 solutions Answer 146823 by jim_thompson5910(28593)   on 2009-05-11 11:40:02 (Show Source): You can put this solution on YOUR website! Start with the given expression. Multiply the first fraction by the reciprocal of the second fraction . Factor to get . Factor to get . Factor to get . Factor to get . Combine the fractions. Highlight the common terms. Cancel out the common terms. Simplify. FOIL Distribute So simplifies to . In other words, where , , or

Polynomials-and-rational-expressions/195819: Please help me with the following problem. I need to express the following rational expression in lowest terms: 3x^2+18x-48 ----------- 2x+16 1 solutions Answer 146821 by jim_thompson5910(28593)   on 2009-05-11 11:34:53 (Show Source): You can put this solution on YOUR website! Start with the given expression. Factor to get . Factor to get . Highlight the common terms. Cancel out the common terms. Simplify. Distribute. So simplifies to . In other words, where

Polynomials-and-rational-expressions/195811: Please help me with the following problem. I need to express the following rational expression in lowest terms: 4x^4-10x^3+8x^2 --------------- 2x 1 solutions Answer 146820 by jim_thompson5910(28593)   on 2009-05-11 11:32:01 (Show Source): You can put this solution on YOUR website! Start with the given expression Factor out from the numerator Highlight the common terms. Cancel out the common terms. Simplify So where

Polynomials-and-rational-expressions/195816: What are the factors of x^2-9x+8? 1 solutions Answer 146819 by jim_thompson5910(28593)   on 2009-05-11 11:26:54 (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 : 1,2,4,8 -1,-2,-4,-8 Note: list the negative of each factor. This will allow us to find all possible combinations. These factors pair up and multiply to . 1*8 2*4 (-1)*(-8) (-2)*(-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 8 1+8=9 2 4 2+4=6 -1 -8 -1+(-8)=-9 -2 -4 -2+(-4)=-6 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).

Polynomials-and-rational-expressions/195817: What are the factors of k^2-6k+5? 1 solutions Answer 146818 by jim_thompson5910(28593)   on 2009-05-11 11:26:24 (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 : 1,5 -1,-5 Note: list the negative of each factor. This will allow us to find all possible combinations. These factors pair up and multiply to . 1*5 (-1)*(-5) 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 5 1+5=6 -1 -5 -1+(-5)=-6 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).

Polynomials-and-rational-expressions/195818: What are the factors of n^2+16n-36? 1 solutions Answer 146817 by jim_thompson5910(28593)   on 2009-05-11 11:25:49 (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 : 1,2,3,4,6,9,12,18,36 -1,-2,-3,-4,-6,-9,-12,-18,-36 Note: list the negative of each factor. This will allow us to find all possible combinations. These factors pair up and multiply to . 1*(-36) 2*(-18) 3*(-12) 4*(-9) 6*(-6) (-1)*(36) (-2)*(18) (-3)*(12) (-4)*(9) (-6)*(6) 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 -36 1+(-36)=-35 2 -18 2+(-18)=-16 3 -12 3+(-12)=-9 4 -9 4+(-9)=-5 6 -6 6+(-6)=0 -1 36 -1+36=35 -2 18 -2+18=16 -3 12 -3+12=9 -4 9 -4+9=5 -6 6 -6+6=0 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).

Linear-equations/195754: Find an equation for the line containing the point (-2,6) and perpendicular to the line 2x + y = 5. 1 solutions Answer 146793 by jim_thompson5910(28593)   on 2009-05-10 23:36:25 (Show Source): You can put this solution on YOUR website! Start with the given equation. Subtract 2x from both sides. Rearrange the terms. We can see that the equation has a slope and a y-intercept . Now to find the slope of the perpendicular line, simply flip the slope to get . Now change the sign to get . So the perpendicular slope is . Now let's use the point slope formula to find the equation of the perpendicular line by plugging in the slope and the coordinates of the given point . Start with the point slope formula Plug in , , and Rewrite as Distribute Multiply Add 6 to both sides. Combine like terms. So the equation of the line perpendicular to that goes through the point is . Here's a graph to visually verify our answer: Graph of the original equation (red) and the perpendicular line (green) through the point .

Complex_Numbers/195737: Multiply: (1/6 – i)^2 1 solutions Answer 146787 by jim_thompson5910(28593)   on 2009-05-10 22:55:58 (Show Source): You can put this solution on YOUR website! Start with the given expression. Expand. Remember something like . Now let's FOIL the expression. Remember, when you FOIL an expression, you follow this procedure: Multiply the First terms:. Multiply the Outer terms:. Multiply the Inner terms:. Multiply the Last terms:. --------------------------------------------------- So we have the terms: , , , and Now add every term listed above to make a single expression. Now combine like terms. So FOILs to . In other words, .

Complex_Numbers/195738: Divide: (6 + 3i)/6 – 3i) 1 solutions Answer 146786 by jim_thompson5910(28593)   on 2009-05-10 22:52:16 (Show Source): You can put this solution on YOUR website! Start with the given expression. Multiply the fraction by . Combine the fractions. FOIL the numerator. FOIL the denominator. Multiply. Combine like terms. Break up the fraction. Reduce. So . So the expression is now in standard form where and