Assignment, due Monday 10/30/23: This week’s video comes from Po-Shen Loh, a math professor at Carnegie Mellon University who, in 2019, discovered a method for factoring quadratics that he had not seen before. Quadratic equations are among the most studied equations in the history of mathematics, and it is very unusual for someone to discover something new about them! Loh published an academic article describing his work, and appeared in a number of popular magazine and newspaper articles (a rare occurrence of mathematical discovery making the news).
Watch the 8-minute portion of the video below (from 6:46 to 14:12), in which Loh explains his method by factoring $x^2-7x+12$.
Test your knowledge by using his method to factor $x^2-8x+13$. (Hint: if you’re stuck, you can watch the next section of the same video (from 14:12 to 18:01) in which Loh goes through this example step-by-step – or check out the section called “Example of Use: a Quadratic That Can’t Be Factored Easily” on Loh’s web page on the topic for yet another example).
Then answer the questions below.
- Who do you think is the intended audience for this video? Do you think the video does a good job of presenting mathematical ideas to its intended audience?
- What do you think is the main idea of Loh’s method, compared with traditional factoring methods?
- Give an example of factoring a quadratic using Loh’s method (you may not use one of the examples from Loh’s video or his website – make up a new one), and explain each step.
- 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.
Sum = -18 => (-9 + u) + (-9 – u) = 18
Product = 32 => (-9 + u)(-9 – u) => 81 – u^2 = 32 => u^2 = 49 => u = -7, 7
Roots: 2, -16
Numbered every question as 1, 2, 3 but it seems like they were all replaced with a, b, c instead (Not a big issue)
I personally believe the intended audience is to students who are just learning how to factor. These students may be looking for extra practice, or simply something to explain to them factoring simpler because they may be struggling. I think the video does a good job at explaining and trying to help students the idea of factoring.
The main idea is basically trying to teach students how to factor and its way around it. He goes into detail and its a great way to help students struggling. I actually agree with Jason, the way he is explaining it to students is the way I was also taught how to factor. The number being used to represent as c is the product and the b is the sum. Till this day!
x^2-5x+4
I believe the intended audience are students like myself and educators who love deeper analysis of concepts. He did a great job explaining the different way to factor that was discovered to students like me and many adults who are probably proficient in quadratic functions. Math smart people. But to a high school student no.
The main idea was to start with a different approach to factoring. Factoring by just using information from the sum. The common way is using the quadratic formula and the one you see what two factors of the constant adds up to the exact co-efficient of the 1st degree term. The difference is that we get an more explicit answer with the common ways. This way in the video seem more confusing and complicated based off the different details to teach someone new to the concept. Its a trick that seems magical because why does it work?
The new way of factoring:
(x^2) + 3x + 2
I believe this video is intended for students who are less algebraically inept. Factoring and isolating variables is much better for students with a strong understanding of such concepts.
I believe this is another method to solve the same problem. For me, if I were a student, these would be extra steps that I would instead not take. Still, from the standpoint of understanding why, finding the common factors of the last term, and then manipulating to find the sum or difference of the middle term works, this is a fantastic method.
Example x^2+6x-16
We have two possibilities for u
3-u, 3+u
Expanding the product of 6-u and 6+u
(3-u)(3+u)=9-u^2
9-u^2=-16
-u^2=-16-9
-u^2=-25
u=5
from eq 1: 3-u
u=-2
from eq 2: 3+u
u=8
(x-2)(x+8)
a) I believe the audience is intended for students in a regents level course and teachers. Factoring is one of those topics that lots of students struggle with remembering what exactly to do so it’s nice to make efforts to simplify it.
b) The difference with Loh’s method as opposed to the traditional way is to start with the sum instead of the product. He breaks down the sum into two parts and introduces a u as something to solve for. From there the expressions are set equal to the product and you can solve for u. Once you get the value of u then you can use the expressions generated in the beginning to find the roots. The traditional way was to find the two numbers that multiplied to the product and added up to whatever the middle term was.
c) x^2+10x-16
(5-u)(5+u)=25-u^2
25-u^2=16
u^2=9
u=3
So the roots are x=2 and x=8
I agree with Irina that this method is simpler because students like us even struggled with factoring so this makes it much easier. Loh did a great job with find this technique because even with numbers that can’t be factored could easily be done with this simple math. I loved Loh’s method and I encourage students to use this.
Taspia Jannat: I think Loh’s intended audience for this video is both the teacher and the student because the teacher can teach using this method and the student can use this new method to factor. I think the video does a good job of presenting mathematical ideas to its intended audience because the way he used a new idea by using the sum first to find the factors is very interesting and much easier even if it requires using a little math. I think the way he started to explain in the beginning was a little confusing. I think this is best when there is a prime number because instead of using the quadratic equation to find the roots, you can easily find it by doing some simple arithmetic. I think the main idea of Loh’s method compared to the traditional factoring method is to make it more easier and simple rather than doing multiple guess and checks. This is more convenient in my opinion and I would definitely use this in the future and I think all teachers should teach this. One example using Loh’s method is x^2-16x+32. Using his steps to solve would look like:
x^2-16x+32=0
x^2-16x+32=(x-___)(x-___)
If we can find 2 numbers with:
sum= 16 and product =32
Then these are all the solutions
So we need u: 8-u, 8+u give product
64-u^2=32
32=u^2
6=u
8-6, 8+6
roots: 2,14
1. I think the intended audience for this video is for all students who have factorization and solving quadratics equations in their courses of algebra, and for anyone else interested in this new approach of factoring quadratics equations. This video presents good mathematical ideas and procedures to the audience in new way of doing so.
2. The main idea of Loh’s method is to show the audience that they can use the traditional factoring methods in reverse order and adding a few techniques to come up to a convenient result of factoring quadratics equations than using the quadratics equations formula, completing square method, or the other way around of his new way of factoring.
3. Factor the quadratic equation:
A: X^2 – 8x + 12 = (x – …)(x – …)
Let find 2 numbers such that:
The sum = 8 ( ignore the – sign of 8) and
The product = 12
We need to find the average value of 8( that is 8/2 = 4), then subtract and add u to that average in form of 2 factors that equalize the product.
B: (4 – u)(4 + u) = 12
16 – u^2 = 12 by the distribution property
4 = u^2 by isolating u^2
2 = u by the square root property
(4 -2 )(4 + 2) = 12 by substituting u equal to 2 from equation B to find the 2 roots or solutions
(2)(6) = 12
X^2 – 8x + 12 = (x – 2)(x – 6) by filling the 2 and 6 in the blank of equation A to the factorization form with roots x = 2 and x = 6.
4. I agree with Irina “ The difference with Loh’s method as opposed to the traditional way is to start with the sum instead of the product.” Because in the form of ax^2 + bx + c in the traditional way; to factor it; is this way called the ac method (meaning finding 2 numbers that their product gives ac and add up to b).
Yadira R. Vazquez