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Quadratic_Equations/151430: . . . . 1 solutions Answer 111318 by jim_thompson5910(28717)   on 2008-08-09 23:34:11 (Show Source): You can put this solution on YOUR website! I'll do the first two to get you started. # 1 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 Subtract from to get Multiply and to get . Take the square root of to get . or Break up the expression. or Break up the fraction for each case. or Reduce. or Simplify. So our answers are or # 2 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 . Simplify the square root (note: If you need help with simplifying square roots, check out this solver) or Break up the expression. or Reduce. So the answers are or which approximate to or

Quadratic_Equations/151432: Will the quadratic equation work on an equation such as x^2 - 3=0? 1 solutions Answer 111317 by jim_thompson5910(28717)   on 2008-08-09 23:29:06 (Show Source): You can put this solution on YOUR website! Start with the given equation. Take note that really looks like 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 Square to get . Multiply to get Rewrite as Add to to get Multiply and to get . Simplify the square root (note: If you need help with simplifying square roots, check out this solver) Break up the fraction. Reduce. or Break up the expression. So our answers are or which approximate to or Note: even though the quadratic formula works perfectly here, there is a much faster way to solve this problem.

Geometry_Word_Problems/151402: Simplify: 8[a – b(3- 4a)] – 6a[2-a(3-b)] Thank you so much for your help. 1 solutions Answer 111295 by jim_thompson5910(28717)   on 2008-08-09 16:51:58 (Show Source): You can put this solution on YOUR website! 8[a – b(3- 4a)] – 6a[2-a(3-b)] = 8[a – 3b+ 4ab] – 6a[2-3a+ab] = [8a – 24b+32ab] – [12a-18a^2+6a^2b] = 8a – 24b+32ab – 12a+18a^2-6a^2b = –24b+32ab-4a+18a^2-6a^2b

Quadratic_Equations/151404: How would I go about writing the quadratic function, in (standard) form, which has a vertex of (-6, 2) and passes through (5, 12)? The standard form is y= a(x-h)^2+k , so i would put -6 as h and 2 as k but from there I am stuck. 1 solutions Answer 111294 by jim_thompson5910(28717)   on 2008-08-09 16:45:47 (Show Source): You can put this solution on YOUR website! Start with the given equation. Plug in and Rewrite as Since the quadratic passes through (5, 12), this means that if you plug in , you will get . So we can plug in and to solve for "a" Plug in and Add Square 11 to get 121 Rearrange the terms Subtract 2 from both sides. Divide both sides by 121. So the value of "a" is This means that the equation is

expressions/151406: Multiply: ((x^2+x-12)/(x^2-x-20))X((x^2+2x-35)/(x^2+9x+14)) 1 solutions Answer 111293 by jim_thompson5910(28717)   on 2008-08-09 16:39:48 (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

Trigonometry-basics/151392: tan(pi/6 + pi/4) 1 solutions Answer 111290 by jim_thompson5910(28717)   on 2008-08-09 15:26:19 (Show Source): You can put this solution on YOUR website! Start with the given expression. Use the identity to expand the expression. Using the unit circle, evaluate to get and evaluate to get Multiply Combine the fractions. Multiply the first fraction by the reciprocal of the second fraction. Cancel out the common terms. Simplify. Multiply the fraction by Combine the fractions. FOIL Combine like terms Factor out the GCF Reduce So

Miscellaneous_Word_Problems/151389: The sum of two numbers is 41 and their difference is 14. Find the numbers. not sure but 2 + x - 14 = 41 1 solutions Answer 111288 by jim_thompson5910(28717)   on 2008-08-09 15:12:28 (Show Source): You can put this solution on YOUR website! "The sum of two numbers is 41" translates to and "their difference is 14" translates to So we have the 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. Notice how the y terms cancel out. Simplify. Divide both sides by to isolate . ------------------------------------------------------------------ Now go back to the first equation. Plug in . Multiply both sides by the LCD to clear any fractions. Distribute and multiply. Subtract from both sides. Combine like terms on the right side. Divide both sides by to isolate . So our answer is and . So the two numbers are and

Expressions-with-variables/151383: ((a/(x+y))+(m/y))/(x/(a+m)) 1 solutions Answer 111281 by jim_thompson5910(28717)   on 2008-08-09 13:47:37 (Show Source): You can put this solution on YOUR website! Start with the given expression. Multiply every term by the LCD to clear the inner denominators. Simplify Rearrange the terms. FOIL Distribute Combine like terms. ---------------------------------------------------- Answer: So simplifies to

Polynomials-and-rational-expressions/151353: simplify: -16x^7y5 / -4x^2y^6 1 solutions Answer 111279 by jim_thompson5910(28717)   on 2008-08-09 13:38:06 (Show Source): You can put this solution on YOUR website! Start with the given expression. Expand. Remember, and Highlight the common terms. Cancel out the common terms. Simplify. Regroup. So simplifies to . In other words, where or

Polynomials-and-rational-expressions/151382: divide x^3-y^3 by x-y 1 solutions Answer 111278 by jim_thompson5910(28717)   on 2008-08-09 13:33:26 (Show Source): You can put this solution on YOUR website! Start with the given expression. Factor to get Highlight the common terms. Cancel out the common terms. Simplify. So simplifies to In other words, where

Graphs/151376: Graph y=2x-1 1 solutions Answer 111267 by jim_thompson5910(28717)   on 2008-08-09 12:26:50 (Show Source): You can put this solution on YOUR website! In order to graph this equation, we only need two points to create a straight line --------------------------------Let's find the first point-------------------------------- Start with the given equation Plug in (you can start with any number, but zero is the easiest number to work with since anything multiplied by zero is zero) Multiply and to get Subtract So when , we have the value . This means we have the first point --------------------------------Let's find the second point-------------------------------- Start with the given equation Plug in Multiply and to get Subtract So when , we have the value . This means we have the second point ------------------------------------------------------------------------------------------------ So we have the two points: and Now plot these two points on a coordinate system Now draw a straight line through the two points. This line is the graph of Graph of through the two points and

Polynomials-and-rational-expressions/151358: factorize 5(3x+y)^2+6(3x+y)-8 please help me understand how to solve this sum. thankyou 1 solutions Answer 111263 by jim_thompson5910(28717)   on 2008-08-09 11:39:29 (Show Source): You can put this solution on YOUR website! Let So the expression goes from to 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 : 1,2,4,5,8,10,20,40 -1,-2,-4,-5,-8,-10,-20,-40 Note: list the negative of each factor. This will allow us to find all possible combinations. These factors pair up and multiply to . 1*(-40) 2*(-20) 4*(-10) 5*(-8) (-1)*(40) (-2)*(20) (-4)*(10) (-5)*(8) 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 -40 1+(-40)=-39 2 -20 2+(-20)=-18 4 -10 4+(-10)=-6 5 -8 5+(-8)=-3 -1 40 -1+40=39 -2 20 -2+20=18 -4 10 -4+10=6 -5 8 -5+8=3 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 Plug in Distribute --------------------------------------------- Answer: So factors to . In other words, Note: you can check the answer by expanding to get .

Graphs/151369: graph of the line through (3, 2) with slope 1/2 1 solutions Answer 111261 by jim_thompson5910(28717)   on 2008-08-09 11:33:43 (Show Source): You can put this solution on YOUR website! If you want to find the equation of line with a given a slope of which goes through the point (,), you can simply use the point-slope formula to find the equation: ---Point-Slope Formula--- where is the slope, and is the given point So lets use the Point-Slope Formula to find the equation of the line Plug in , , and (these values are given) Distribute Multiply and to get Add 2 to both sides to isolate y Combine like terms and to get (note: if you need help with combining fractions, check out this solver) ------------------------------------------------------------------------------------------------------------ Answer: So the equation of the line with a slope of which goes through the point (,) is: which is now in form where the slope is and the y-intercept is Notice if we graph the equation and plot the point (,), we get (note: if you need help with graphing, check out this solver) Graph of through the point (,) and we can see that the point lies on the line. Since we know the equation has a slope of and goes through the point (,), this verifies our answer.

Graphs/151365: graph linear equation y = 2x - 1 1 solutions Answer 111258 by jim_thompson5910(28717)   on 2008-08-09 11:31:07 (Show Source): You can put this solution on YOUR website! Looking at we can see that the equation is in slope-intercept form where the slope is and the y-intercept is Since this tells us that the y-intercept is .Remember the y-intercept is the point where the graph intersects with the y-axis So we have one point Now since the slope is comprised of the "rise" over the "run" this means Also, because the slope is , this means: which shows us that the rise is 2 and the run is 1. This means that to go from point to point, we can go up 2 and over 1 So starting at , go up 2 units and to the right 1 unit to get to the next point Now draw a line through these points to graph So this is the graph of through the points and

Graphs/151372: write the equation of y = 1/3x - 2 in standard form using only integers and a positive coefficient of x 1 solutions Answer 111257 by jim_thompson5910(28717)   on 2008-08-09 11:21:26 (Show Source): You can put this solution on YOUR website! Start with the given equation. Multiply both sides by 3 to eliminate the fractions. Distribute and multiply. Subtract from both sides. Rearrange the terms. Multiply both sides by -1. This will make the x coefficient positive. Distribute and multiply. So the equation is now in standard form where , and

Graphs/151371: 4. Find the slope of the line that contains the points (-11,5) and (5,-8) 1 solutions Answer 111256 by jim_thompson5910(28717)   on 2008-08-09 11:16:01 (Show Source): You can put this solution on YOUR website! Start with the slope formula. Plug in , , , , , Subtract from to get Subtract from to get So the slope of the line that goes through the points and is

Radicals/151325: 5 square root of x + 1 = 3 2/3 = 2 - 5x-3 over x-1 solve algebraically thanks 1 solutions Answer 111241 by jim_thompson5910(28717)   on 2008-08-08 21:53:58 (Show Source): You can put this solution on YOUR website! I'll do the first one to get you started. Start with the given equation. Divide both sides by 5. Square both sides. Square to get Subtract 1 from both sides. Combine the fractions. So the solution is which approximates to

Matrices-and-determiminant/151324: I am lost with this assignment, HELP! I do not understand hwo these are answered! 4. A group of students decides to sell pizzas to help raise money for their senior class trip. They sold pepperoni for $12, sausage for $10, and cheese for $8. At the end of their sales the class sold a total of 600 pizzas and made $5900. The students sold 175 more cheese pizzas than sausage pizzas. Set up a system of three equations and three unknowns, use an augmented matrix to solve, and show all work to receive full credit. A. What are the three unknowns? B. Write a separate equation representing each of the first three sentences. C. Determine the augmented matrix that represents the three equations. D. Solve for the matrix. Show all work to receive full credit. E. How many of each type of pizza were sold? 1 solutions Answer 111240 by jim_thompson5910(28717)   on 2008-08-08 21:50:26 (Show Source): You can put this solution on YOUR website! a) Let x=# of pepperoni sold, y=# of sausage sold, and z=# of cheese sold b) Since there are 3 unknowns, we need 3 equations: Equation #1: First, since they sold "pepperoni for $12, sausage for $10, and cheese for $8" and collected $5,900, this means that Equation #2: Also, we know that the students sold 600 pizzas total. So this means that the sum of all of the pizzas sold was 600. So this translates to . Equation #3: Finally, since they sold "175 more cheese pizzas than sausage pizzas", this makes the equation Getting all of the variables to one side for equation 3 gives us: So we have the system c) This means that we have the augmented matrix matrix d) Solution was generated by the Linear Algebra Toolkit ============================================================= Answer: So the solutions are , , and (these values form the right hand column of the augmented matrix). So this means that they sold 225 pepperoni pizzas, 100 sausage pizzas, and 275 cheese pizzas.

Linear-systems/151320: Hello may someone please elp me with this question im very confused! The length of a rectangle is 3 in. more than the width. The perimeter of the rectangle is 24 in. I THINK THAT I AHVE FOUND THE VARIABLES LENGTH=3iN. WIDTH=24iN.(AM I RIGHT???) What is the expression for the length in terms of the width? What is the length of the rectangle? 1 solutions Answer 111236 by jim_thompson5910(28717)   on 2008-08-08 21:26:58 (Show Source): You can put this solution on YOUR website! Let x=width and y=length Since the "length of a rectangle is 3 in. more than the width", this means that . Also, because the "perimeter of the rectangle is 24 in", this tells us that Start with the formula for the perimeter of a rectangle Plug in and Distribute Subtract 6 from both sides. Combine like terms. Divide both sides by 4. Reduce. So the first part of the answer is or . This means that the width is 4.5 inches. Go back to the first equation Plug in Combine the fractions. So the second part of the answer is or . This means that the length is 7.5 inches. ======================================================= Answer: So the width is 4.5 inches and the length is 7.5 inches

Quadratic-relations-and-conic-sections/151316: Please help me divide and simplify: (3x^4 - 2x^3 + 16x - 192) / (x^2 - 8) 1 solutions Answer 111234 by jim_thompson5910(28717)   on 2008-08-08 21:18:08 (Show Source): You can put this solution on YOUR website! This solution was generated from the Long Division Calculator So

Distributive-associative-commutative-properties/151323: how would you... A.) use the distributive property to solve and create a binomial to set up the FOIL method B.) Solve with the FOIL method could you break it down step by step? 4x(x^2-2x+5) 1 solutions Answer 111233 by jim_thompson5910(28717)   on 2008-08-08 21:13:22 (Show Source): You can put this solution on YOUR website! This problem does not involve FOILing, but involves distribution (you use FOIL when you have two binomials). So I would double check the problem. Anyways, here's how you would simplify the problem. Start with the given expression. Distribute. Multiply and to get Multiply and to get Multiply and to get So distributes and simplifies to In other words,

Functions/151318: y-2-8 sqrt(y-2)+15=0 1 solutions Answer 111230 by jim_thompson5910(28717)   on 2008-08-08 21:08:49 (Show Source): You can put this solution on YOUR website! Start with the given equation. Combine like terms. Subtract 13 from both sides. Subtract from both sides. Square both sides. FOIL Distribute Subtract from both sides. Add 128 to both sides. Combine like terms 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 Subtract from 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 our answers are or

decimal-numbers/151321: Write the solution using interval notation. x<-6 or x>0 1 solutions Answer 111229 by jim_thompson5910(28717)   on 2008-08-08 21:03:35 (Show Source): You can put this solution on YOUR website! The solution in interval notation is: () () Also, the solution in set notation is: Here's the graph of the solution set Note: There is an open circle at which means that we're excluding that value from the solution set. Also, there is an open circle at which means that we're excluding that value from the solution set.

Polynomials-and-rational-expressions/151301: how do I simplify: (-4t^2p^4u^6)^4 1 solutions Answer 111211 by jim_thompson5910(28717)   on 2008-08-08 16:19:06 (Show Source): You can put this solution on YOUR website! Start with the given expression. Rewrite as Distribute the outer exponent to the inner exponents Multiply the exponents. Raise -4 to the 4th power to get 256 So .

Polynomials-and-rational-expressions/151305: simplify: (-5r^2t^5u^6)^3 1 solutions Answer 111210 by jim_thompson5910(28717)   on 2008-08-08 16:15:12 (Show Source): You can put this solution on YOUR website! Start with the given expression. Rewrite as Distribute the outer exponent to the inner exponents Multiply the exponents. Cube -5 to get -125 So .

Systems-of-equations/151299: Could you help solve: 8/x-5/y=51/14 6/x+(-10)/y=-(29/7) 1 solutions Answer 111201 by jim_thompson5910(28717)   on 2008-08-08 14:43:54 (Show Source): You can put this solution on YOUR website! Start with the first equation. Multiply both sides by . This will eliminate the fractions. Distribute and multiply Subtract from both sides. Factor out from the right side. Divide both sides by . So after isolating y of the first equation, we get --------------------------------- Move onto the second equation Plug in . In other words, replace "y" with Multiply the numerator "-10" by the reciprocal of the fraction in the denominator. Distribute Simplify. Multiply both sides by . This will eliminate the fractions. Simplify Subtract from both sides. Combine like terms. Divide both sides by -800. Reduce. So this is the first part of the answer. Go back to the first isolated equation Plug in Multiply and reduce Subtract Multiply the first fraction by the reciprocal of the second fraction. Multiply and reduce ========================================= Answer: So the answer is and which forms the ordered pair

Linear-systems/151298: Could you please help me solve the system of equations? 3x+3y+z=5 x-3y+2z=-13 8x-2y+3z=-8 1 solutions Answer 111200 by jim_thompson5910(28717)   on 2008-08-08 14:29:32 (Show Source): You can put this solution on YOUR website! Add equations 1 and 2. Let's call the result equation 4: + ----------------- Equation 4: Multiply both sides of equation 1 by 2 Multiply. Let's call this equation 5 ------- Multiply both sides of equation 3 by 3 Multiply. Let's call this equation 6 Now add equations 5 and 6. Let's call the result equation 7: + ------------------- Equation 7: =================================== So we now have the two equations equation 4 and equation 7: equation 4: equation 7: Start with the given system of equations: Now in order to solve this system by using substitution, we need to solve (or isolate) one variable. I'm going to solve for z. So let's isolate z in the first equation Start with the first equation Subtract from both sides Rearrange the equation Divide both sides by Break up the fraction Reduce --------------------- Since , we can now replace each in the second equation with to solve for Plug in into the first equation. In other words, replace each with . Notice we've eliminated the variables. So we now have a simple equation with one unknown. Distribute to Multiply Multiply both sides by the LCM of 3. This will eliminate the fractions (note: if you need help with finding the LCM, check out this solver) Distribute and multiply the LCM to each side Combine like terms on the left side Add 88 to both sides Combine like terms on the right side Divide both sides by 46 to isolate x Divide So the first part of our answer is: Since we know that we can plug it into the equation (remember we previously solved for in the first equation). Start with the equation where was previously isolated. Plug in Multiply Combine like terms and reduce. (note: if you need help with fractions, check out this solver) So the third part of our answer is: Go back to the first equation Plug in and Multiply Combine like terms. Add to both sides. Combine like terms on the right side. Divide both sides by to isolate . Reduce. So the second part of the answer is -----------------Summary------------------------------ So our answers are: , , and They form the point (1,2,-4)

Systems-of-equations/151297: Can you help me solve this system algabraically and support the solutions graphically? y+x^2 9x-y=20 (ordered pairs) 1 solutions Answer 111199 by jim_thompson5910(28717)   on 2008-08-08 14:19:43 (Show Source): You can put this solution on YOUR website! What is equal to? This is not an equation. Did you mean to say ?? Start with the second equation. Plug in Subtract 20 from both sides. Rearrange the 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 Square to get . Multiply to get Subtract from 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 x answers are or Go back to the first equation Plug in Square 4 to get 16. So the first set of answers is and which form the ordered pair (4,16) ---------------- Go back to the first equation Plug in Square 5 to get 25. So the second set of answers is and which form the ordered pair (5,25) ========================================== Answer: So the ordered pair solutions are (4,16) and (5,25) Notice that if we graph the two equations, we get Graph of (red) and (green). It's hard to see, but with the help of a graphing calculator, we can see that the graphs intersect at the points (4,16) and (5,25). So this verifies our answer.

Polynomials-and-rational-expressions/151296: u^3/w ÷ uv^3/w^2 1 solutions Answer 111197 by jim_thompson5910(28717)   on 2008-08-08 13:56:23 (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. Combine the fractions. Multiply Expand. Remember, and Highlight the common terms. Cancel out the common terms. Simplify. Regroup. So simplifies to . In other words, where , , or

Exponential-and-logarithmic-functions/151295: - How many months does it take for $1000 to grow to $1500 in an account paying 12% compounded monthly? 1 solutions Answer 111196 by jim_thompson5910(28717)   on 2008-08-08 13:52:24 (Show Source): You can put this solution on YOUR website! Start with the compound interest formula. Plug in , , (note: 12%=0.12) and Divide both sides by 1000. Reduce Divide to get 0.01 Add 1 to 0.01 to get 1.01 Take the log of both sides. Rewrite the right side using the identity Divide both sides by . So the answer is which approximates to So it takes about 3.4 years (or 40.8 months) to go from $1,000 to $1,500

Polynomials-and-rational-expressions/151294: 3x^2 + 8x - 3 = 0 1 solutions Answer 111195 by jim_thompson5910(28717)   on 2008-08-08 13:43:40 (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 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 our answers are or