Assignment, due Monday, 3/14/22: This week’s reading comes from a New York Times article This Professor’s ‘Amazing’ Trick Makes Quadratic Equations Easier. Did you know that, as a CUNY student, you have free access to the New York Times online? Instructions for accessing the New York Times can be found by following the link below – use the instructions to sign up for your free account, and read the article.
NOTE: if you have trouble with the online version, you can view a pdf of the article here:
- This article is somewhat unusual, in that it presents mathematical details in a very general-interest, widely-read platform (the New York Times). Who do you think is the intended audience for this article? Do you think the article does a good job of presenting mathematical ideas to its intended audience?
- Carefully work through the section “How the new method works” to learn Dr. Loh’s method for find roots/factoring quadratics. Describe in your own words how the method works, by giving an example of a quadratic function and describing the steps of factoring it using his method.
- Extra Credit. Respond to one of your fellow students’ comments. Do you agree? Disagree? Did their comments make you think or provoke additional questions? Reminder: Be respectful, be kind.
I think the intended audience for this article is middle and high school students who have studied mathematics, and it is difficult to understand for those who have never studied mathematics. I think Dr.Lu did a very good job of showing how to factor because he used the meaning of sum, which is a very well understood concept for the human brain.
x^2+ax+b, we can step for a first which is a/2-u and a/2+u and we can get and we have to make (a/2-u )(a/2+u) =c, after we can get [(a/2)^2-u^2]=c, then we can get two numbers for u, and we put the two number back to a/2-u to get two number, which is the roots for the function.
I really like this method to find the roots of the quadratic function. I think this method is very simple and easy to understand and can reduce the possibility of error. If I become a teacher in the future I will definitely share this method with my students.
I mean the c is the b!
I am not used to using this method to solve quadratic equations. I preferred the traditional way to do it. However, I should learn them if this method is easy for students to learn.
I disagree with your example. You omitted the ‘-‘ sign for a/2 for roots because the axis of symmetry is -a/2.
1. The audience for this article is math teachers who teach quadratic equations. This article may arise the curiosity of a new method to solve quadratic equations. However, I would like to see if this method really works and is more effective than the traditional method.
2. Dr. Loh’s method to find roots starts from computing the midpoint of two roots. First, we compute the multiply (-1/2) to b from “ax^2+bx+c”. Then we express two roots by adding/subtracting the variable(unknown) from the midpoint. Finally, we compute two roots by multiplying two roots and set the equation that equals to c from “ax^2+bx+c”.
Example: y= x^2+6x+6
1) (-1/2)*b=-3 (midpoint)
2) r_1=-3-u, r_2=-3+u
3) r_1*r_2=(-3-u)(-3+u)=6
9-u^2=6
u=+-sqrt(2) then r_1=-3-sqrt(2), r_2=-3+sqrt(2)
1) Generally speaking newspapers are written at a eighth grade reading level. The fact that the discussion about applications was presented comes off as convincing people why quadratic equations are important. This points to the ideal audience being people who have or are currently going through algebra but do not fully understand or are interested in algebra. Math has a massive image problem so articles such as these can be helpful in reclaiming the narrative but no the only stragey.
2) The method used by Dr Loh uses the midpoint of a quadratic function let’s take y=x^2-6x-8. First divide the middle term by 2, so in this case 6/2=3. You will have two parts to the left and the right of the value you found (midpoint). To express that you can write this as r_1=3-u and r_2=3+u. To be able to use the equations we need to multiply r_1 and r_2, which gives us
(3-u)(3+u)=9-u^2. To solve for u we need to set 9-u^2 equal to the last term of our original equation. So this is what we end up with 9-u^2=8 so when solve for u you get u=+-sqrt(17), then r_1=3-sqrt(17), r_2=3+sqrt(17).
I agree with you that math has massive image problems but specifically, this specific method didn’t help me to understand the quadratic equation better, But it’s always good to know more than one problem-solving strategy.
I think this article is geared towards enthusiasts and maybe teachers. NYT is a broad platform, so they want to reach a broad platform. As confusing and narrow as math is, difficulty still sparks curiosity. Dr. Loh uses the midpoint of two roots. Let’s take y=x^2+18x+9. You can simply this expression by dividing the second term by three. Now you have two equations to work with. x1=6+x and x2=6-x
Using factoring of perfect squares we get (x+6)(x-6)=x^2-36 set the factored equation equal to the last term we do aokw algebra and get
x^2-36=9
x^2=+/-sqrt(45)
This is a really different way to think about factoring where they analyze the graph carefully in order to solve the problem. I think this article was designed for people who want to learn a method way to factor. I also believe this article was created for mathematics enthusiasts who want to learn math because its fun not because it requires it. For me it wasn’t the best way to learn to factor, however, many people might like this method.
first of all this method has uses the fact that parabolas are symmetrical from the axis of symmetry.
For example, we can consider the function Y= x^2+5x+6
suppose R and S are two roots of the function
we want R+S=-b which is -5
we divide -5/2
therefore,
according to this method
R= -5/2-u
S=-5/2+u
we want the product of R and S equal to c which is 6
(-5/2-u)(-5/2+u)=6
u= 1/2 after calculation
R=-2.5-.5=-3
S=-2.5+.5=-2
therefore,
X=-3 and X=-2
or Y= (X+3)(X+2)