Solving equations by factoring

We looked at mostly quadratic equations, but also one cubic equation, to see how factoring could be used to solve them using the zero factor property (ZFP). That is, if ab=0, the either a or b must be zero.

We also looked at how the TI can be used (graphically, and with TABLE) to check our answers. Great class all, you were all great today! HW is 331: 3-17

Solving equations by factoring

Work Problem 5

The following was a question a anonymous visitor asked: Jim can fill a pool carrying buckets of water in 30 minutes. Sue can do the same job in 45 minutes. Tony can do the same job in 1 ½ hours. How quickly can all three fill the pool together?



Work problem 5 solution here

Quadratic Equation Question Answers

Fox 214

http://www.8foxes.com/

Ajit Athle submitted this problem, as it neatly fits in our series on Equilateral Triangles. He said:

"Here's a simple problem to go in your current series on ET's. In equilateral triangle ABC, we've cevians BD (D on AC) & CE (E on AB) such that 2*BE = AE & 2*AD = DC. If CE & BD intersect in P, then prove, w/o using trig! onometry or co-ordinate geometry, that AP is perpendicular to CE."
Thank you Ajit!

Problems on Co-ordinate Geometry

The Trade Off: Risk-Reward vs. Probability of Profit

When entering the realm of options spread trading, it is imperative that a trader understands probability of profit. Not only how to calculate it, but also the role it plays in the decision making process. I would hope most visitors to this blog are already cognizant of this concept, but there are no doubt newcomers to the options arena that might benefit from an overview. In my next two posts I'll review a simple method for calculating probability of profit and illustrate the relationship between increasing probability of profit and decreasing ris! k-reward. Be forewarned, there will be some math involved, so hold your head still so nothing spills out!

As a precursor to calculating probability of profit, a trader must first understand the greek delta. One of the characteristics of delta is it calculates the probability of an option expiring in-the-money. For example, suppose stock XYZ is trading at $100 and the 90 strike put option has a delta of .20. This means there is a 20% probability that the 90 strike put will be in-the-money at expiration. Put another way, there is a 20% probability the stock price will be below $90 at expiration. Now, we can use a little arithmetic to calculate the probability of an option expiring out-of-the-money. We can all agree that there is a 100% probability of the stock price residing somewhere. If there is a 20% chance the stock will be below $90, then it stands to reason that there is an 80% chance of the stock residing above $90 at expiration. Th! us the formula for calculating the probability of an option ex! piring o ut-of-the-money is: 1 minus delta.

Although delta can be used to calculate probability of profit on most option spread trades, I'm going to focus on vertical spreads. Remember, the four verticals are the bull call, bull put, bear call, and bear put spreads. The two bullish spreads consist of buying a lower strike option and selling a higher strike option of the same type in the same expiration month. To realize the maximum profit we want the stock to be above the higher strike price at expiration. Alternatively, the two bearish spreads are constructed by buying a higher strike option and selling a lower strike option of the same type in the same expiration month. Capturing the maximum profit on these two spreads requires the stock to be below the lower strike price at expiration. To calculate the probability of profit on a bull spread, simply use delta to calculate the probability of the stock residing above the higher strike. Conv! ersely, for a bear spread calculate the probability of the stock residing below the lower strike.

Suppose stock ABC is trading at $50 and we enter a bull put spread by simultaneously buying the 40 put and selling the 45 put. To realize our maximum profit we need the stock to be above $45 at expiration. Using delta we can calculate the probability of the stock residing above $45, thereby calculating our probability of profit. The current delta of the 45 put is .30, implying the stock has a 30% probability of residing below 45 at expiration. We can plug this delta (.30) into our formula: 1 - .30 = .70. In addition to knowing the risk-reward of the 45-40 spread, I now know the likelihood of realizing my profit is 70%.

For other posts on delta, check out:

Delta

Delta and Probabilities

Fixed Delta and Risk Graphs

Variable Delta and Risk

Stay tuned for Part Two...

how to calculate probability

DARPA Mathematical challenges

Via Ars Mathematica and Not Even Wrong, a DARPA-issued list of challenges in mathematics for the 21st century. Some that puzzled me:

• Computational Duality: Duality in mathematics has been a profound tool for theoretical understanding. Can it be extended to develop principled computational techniques where duality and geometry are the basis for novel algorithms?
  
  I'm not sure what this is trying to say, or whether I'm reading it wrong, because the story of linear programming, primal dual schemes, and the Lagrangian is the story of using "duality and geometry as the basis for novel algorithms"
• ! What are the Physical Consequences of Perelman’s Proof of Thurston’s Geometrization Theorem?
  Can profound theoretical advances in understanding three-dimensions be applied to construct and manipulate structures across scales to fabricate novel materials?
  
  Thurston's geometrization conjecture talks about the structure of 3-manifolds: i.e 3 dimensional surfaces living in higher dimensional spaces ? What kinds of materials could be fabricated using this ?
  
• Computation at Scale: How can we develop asymptotics for a world with massively many degrees of freedom?
  
  This sounds like there's a nice computational question lurking somewhere, but I'm not quite sure where.
  

Nice to see algor! ithmic origami and self-assembly also mentioned. I am particul! arly int rigued by the reference to the geometry of genome spaces.

linear programming dual

Linear Relationship Open-ended Questions

I'm working on asking more conceptual, open-ended questions in order to challenge students, encourage critical thinking, utilize the "Rule of 4", and prepare students for AP Calculus level of rigor. The questions below represent my first serious attempt. Visit ilovemath.org for the full pdf.

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

1. Give an example of a linear relationship in graphical, numeric, and analytic (equation) forms. Use the same linear relationship for all three representations.
2. What is the relationship between the slope formula and the point-slope form of a line? (How can you derive one from the other?)
3. How can you identify two perpendicular ! lines if they are both in General Form? (What is special about the numbers A, B, and/or C between the two equations)
4. Why is the “intercept form” given its name? (What makes it different from the slope-intercept form?). Also, give an example of a linear relationship in intercept form and graph the line.
5. Will 2 pair of parallel lines that are perpendicular to each other always form a square on their interior? If so, state how you know and if not, provide a counter example.
1. If : L1 || L2 and L3 || L4 , L1 _|_ L3 and L4 , L2 _|_ L3 and L4
2. Then: Does the interior of these lines always form a square?
6. Given the four lines described in question 5, if you multiplied the slopes of the 4 lines together, what would be the product?
7. Do the following four points form a parallelogram? How do you know if it does or does not. Points: (−4,0), (2,4), (−2,−3), (4,1)
8. The table below! gives the price, the supply, and the demand, for a certain vi! deo game .
   
1. Graph the points representing price & supply and the points representing price & demand.
2. Estimate the price at which the supply of video games will equal the demand. Also estimate the quantity that is supplied/demanded at this price.
3. What happens to the supply and to the demand when the price of the video game is higher than the price you found in part b? .... lower than the price of b?
   
   

Linear Functions and Slope Forms

List of Publications

1. N. Dilna, A. Ronto. Unique solvability of a non-linear non-local boundary-value problem for systems of non-linear functional differential equations. Mathematica Slovaca, Vol. 60 (2010), No. 3, pp. 327-338
2. N. Dilna, M. Fečkan. About the uniqueness and stability of symmetric and periodic solutions of weakly nonlinear ordinary differential equations. Dop. Nats. Akad. Nauk Ukrainy, (2009), No. 5, pp. 22- 28 (in Russian).
3. N. Dilna, M. Fečkan. On the uniqueness and stability of symmetric and periodic solutions of weakly nonlinear ordinary differential equations. Miskolc Mathematical Notes. Vol. 10 (2009), No. 1, pp. 11-40. URL: http://mat76.mat.uni-miskolc.hu/~mnotes/contents.php?number=+1+&volume=10
4. N. Dilna and M. Fečkan. Weakly non-linear and symmetric periodic systems at resonance. Journal Nonlinear Studies, Vol. 16 (2009), No. 2, pp. 23-44. URL: www.nonlinearstudies.com/old/journal/Members/vol_16,_no.2,_2009.htm
5. N. Dilna, A. Ronto. General conditions guaranteeing the solvability of the Cauchy problem for functional differential equations. Mathematica Bohemica. Vol. 133 (2008), No. 4, pp. 435-445.
   
6. Nataliya Dilna. On Un! ique Solvability of the Initial Value Problem for Nonlinear Fu! nctional Differential Equations. Memoirs on Differential Equations and Mathematical Physics. Vol. 44 (2008), pp. 45-57. URL: http://www.jeomj.rmi.acnet.ge/memoirs/vol44/contents.htm
7. N. Z. Dilna, A. N. Ronto, V. A. Pylypenko. Some coditions for the unique solvabilityof a nonlocal boundary-value problem for linear functional differential equations. Dop. Nats. Akad. Nauk Ukrainy, (2008), No. 6, pp. 13- 18 (in Ukrainian).
8. A. Ronto, V. Pylypenko, N. Dilna. On the Unique Solvability of a Non-Local Boundary Value Problem for Linear Functional Differential Equations. Mathematical Modelling and Analysis. Vol. 13 (2008), No. 2, pp. 241-! 250. URL: http://inga.vgtu.lt/~art/
9. N. Z. Dilna, A. N. Ronto. General conditions of the unique solvability of the Cauchy problem for systems of nonlinear functional-differential equations. Ukrainian Mathematical Journal. Vol.60 (2008), No. 2, pp. 167-172.
10. A. N. Ronto, N. Z. Dilna. Unique solvability conditions of the initialvalue problem for linear differential equations with argument deviations. Nonlinear Oscillations. Vol. 9 (2006), No. 4, pp. 535-547.
11. A. M. Samoilenko, N. Z. Dilna, and A. N. Ronto. Solvability of the Cauchy problem for linear integral-differential equations with transformed! arguments. Nonlin ear Oscillations. Vol. 8 (2005), No. 3, pp. 388-403.
12. N. Dilna. On the solvability of the Cauchy problem for linear integral differential equations, Miskolc Mathematical Notes. Vol. 5 (2004), No. 2, pp. 161- 171. URL: http://mat76.mat.uni-miskolc.hu/~mnotes/contents.php?volume=5&number=2#article104
13. N. Z. Dilna and A. N. Ronto. On the solvability of the Cauchy problem for systems of linear functional differential equations with (\sigma, \tau)-positive right-hand sides. Dop. Nats. Akad. Nauk Ukrainy, (2004), No. 2, pp. 29- 35 (in Russian).
14. N. Z. Dilna and A. N. Ronto. New solvability conditions for the Cauchy problem! for systems of linear functional differential equations. Ukrainian Mathematical Journal. Vol. 56 (2004), No. 7, pp. 867 - 884.
15. N. Dilnaya and A. Ronto. Multistage iterations and solvability of linear Cauchy problems, Miskolc Mathematical Notes. Vol. 4 (2003), No. 2, pp. 89-102. URL: http://mat76.mat.uni-miskolc.hu/~mnotes/contents.php?volume=4&number=2#article81
    

    Preprints

• Nataliya Dilna, Michal Fečkan. On the uniqueness and stability of symmetric and periodic solutions of ! weakly nonlinear ordinary differential equations. Preprint of the Mathematical Institute of the Slovak Academy of Sciences, Bratislava. 3/2008 (July 8, 2008), 30 p. http://www.mat.savba.sk/preprints/2008.htm
• Nataliya Dilna, Michal Fečkan. Weakly nonlinear and symmetric periodic systems at resonance. Preprint of the Mathematical Institute of the Slovak Academy of Sciences, Bratislava. 1/2009 (February 9, 2009), 21 p. http://www.mat.savba.sk/preprints/2009.htm

Citations

The paper [14] N. Dilnaya and A. Ronto. Multistage iterations and solvability of linear Cauchy problems, Miskolc Mathematical Notes. Vol. 4 (2003), No. 2, pp. 89-102

has been cited in such works:

1. J. Å remr. On the innitial value problem ! for two-dimensional systems of linear functional-differentiona! l equati ons with monotone operators. Preprints of Academy of Sciences of the Czech Republic. 162/2005, 53 p.
2. J. Å remr. A note on two-dimensional systems of linear differential inequalities with argument deviations, Miskolc Mathematical Notes. 7, No. 2, 171-187, 2006, MR, ZBL MATH
3. J. Å remr. On systems of linear functional differential inequalities, Georgian Mathematical Journal. 13(3), pp. 539-572, 2006. MR, ZBL MATH
4. J. Å remr. On the Cauchy type problem for systems of functional-differential equations. Nonlinear Analysis, Theory, Methods and Applications. 67, no. 12, pp. 3240-3260, 2007. SCI
5. J. Å remr and R. Hakl. On the Cauchy problem for two-dimensional systems of linear functional differential equations with monotone operators, Nonlinear Oscillations. 10(4), pp. 560-573, 2007. SCOPUS
6. E. I. Bravyi. On the solvability of the Cauchy problem for systems of two liner functional differential equations. Memoirs on Differential Equations and Mathematical Physics. 41, pp. 11-26, 2007. MR, ZBL MATH
7. J. Å remr. On the Cauchy type problem for two-dimensional functional-differential systems. Memoirs on Differential Equations and Mathematical Physics. 40, pp. 77-134, 2007, MR, ZBL MATH
8. J. Å remr. Solvabiliy conditions of the Cauchy! problem for two-dimensional systems of linear functional-diff! erential equations with monotone operators. Mathematica Bohemica 132(2), 263-295, 2007.
9. J. Sremr. On the initial problem for two-dimensional systems of linear functional-differential equations with monotone operators. Fasciculi Mathematici. Nr 37, pp. 87-108, 2007
10. Z. Oplustil. On constant sign solution (nonpositive) of certain functional differentional inequality. Mathematical models in engineering, biology and medicine. Book Series: AIP Conference Proceedings. 1124, pp. 274-283, 2009, SCI
11. A. Lomtatidze, Z. Opluštil та J. Šremr. Nonpositive solutions to a certain functional differential inequality. Nonlinear Oscillations. 12(4) , pp. 461-494, 2009
12. J. Sremr. On the initial value problem for two-dimensional linear functional differential systems. Memoirs on Differential Equations and Mathematical Physics, 50, pp. 1-127, 2010.
    
    

The paper [7] A. Ronto, V. Pylypenko, N. Dilna. On the Unique Solvability of a Non-Local Boundary Value Problem for Linear Functional Differential Equations. Mathematical Modelling and Analysis. Vol. 13 (! 2008), No. 2, pp. 241-250.

has be en cited in such work:

• Z. OpluÅ¡til, J. Å remr, On a non-local boundary value problem for linear functional differential equations, Electron. J. Qual. Theory Differ. Equ. (2009), No. 36, 1-13.

linear differential equations