The need to solve quadratic equations never seems to disappear. Whether it is completing a physics problem, solving a differential equation, or performing equilibrium calculations in chemistry, quadratic equations are an integral part of all STEM-based disciplines.

Depending on the complexity of the quadratic equation, the typical 'guess-and-check' method taught in most high school classes can often be frustrating and time-consuming. Professor of mathematics Dr. Po-Shen Loh, in his new method shown here, recognizes some important properties of solutions to quadratic equations and integrates them into a more intuitive approach that students are much more likely to feel motivated by.

For example, consider the equation x^2 - 14x + 45 = 0. Most students are taught to first factor this equation by thinking of two numbers that multiply to 45 and add to -14. After trying multiple values, we would discover that those values are -5 and -9. We would use these values to factor the equation into the form (x-5)*(x-9) = 0. Setting each factor equal to zero, we would get x = 5 or x = 9. Equivalently, to solve for x more directly, we need two numbers that multiply to 45 and add to 14 (again, x = 5 and x = 9).

The only way to speed up this process of guess-and-check is to do enough similar problems until the guesses become second nature. Not to mention, this becomes exponentially more difficult as the coefficient on x^2 increases (for example, solving the equation 6x^2 + 7x - 20 = 0).

For the example above, Dr. Loh's method builds on a simple starting point:

(i) We know that the numbers (call them R and S) add to 14

(ii) We know that since the numbers add to 14, they must have a mean value of 14/2 = 7

(iii) If the two numbers have an average of 7, they must be an equal 'distance' (call this distance z) from 7

(iv) We can write the two numbers as R = 7+z and S = 7-z

(v) Since the numbers R and S multiply to 45, then (7+z)*(7-z) = 45 ⇒ 49 - z^2 = 45. In other words, z^2 = 4, so z = +2 or z = -2

(vi) The solution to the equation is then R = 7+2 = 9 and S = 7-2 = 5 (as we predicted)

We can generalize this idea for any complex coefficients a, b and c in the equation ax^2 + bx + c = 0 to actually prove the quadratic formula. However, using Dr. Loh's method on specific examples (as above) helps build intuition for why the quadratic formula works in the first place. Other proof methods such as completing the square are just as mathematically sound, but they do not utilize the mathematical instinct that makes solving a problem in mathematics so gratifying.

Although I am currently a student working for Maplesoft, I had not used Maple Learn extensively beforehand. Dr. Loh's idea of creating a more intuitive way to solve such a conventional problem inspired me to create a document in Maple Learn, linked here, outlining the steps above.

Learning new ways to solve a problem in mathematics is exciting, but it is often difficult to present in a way that is clear, visually-appealing and easy to create. Most online mathematical environments are difficult to navigate and typically lack visualizations to accompany an idea. With Maple Learn, it felt comforting to open a clean canvas where I was able to easily build a document in just a few hours that not only summarized the main ideas of this new method, but also showed the user why the method works using live animations and colour schemes (see some examples below).

I surprised myself (as well as my managers) by how quickly I was able to transfer all of my ideas into the document. I could also split related content into groups and use collapsible sections to keep the document uncluttered and easy to read.

I also took advantage of the freedom to explore other documents and directly reference them through hyperlinks.

Sometimes it can be difficult to follow a new concept without having some background information. Adding these references makes it simple for the reader to access supporting documents and ensure there are no knowledge gaps to be filled along the way. Once you make a document, you also have the option to publish it to your own gallery and make it public for others to use and learn from.

Maple Learn has been incredibly helpful for sharing the things that interest me the most. If you have something related to mathematics that excites you, try not to keep it to yourself. Consider using Maple Learn to share your ideas with the world and see your vision come to life!

A convex polyhedron with 90 vertices recently claimed to be the first known without the Rupert property

A polyhedron is said to have the Rupert property if there is a way to cut a hole through it so that an identical copy of the original polyhedron can pass through it.  For example, you can see that the cube has the Rupert property in the following plot.

P := plots:-polyhedraplot([0,0,0], polytype=hexahedron, style=line, color="DarkBlue", axes=none):
Q :=  plots:-polyhedraplot([0,0,0], polytype=hexahedron, style=line, color="Gold", axes=none):
plots:-display(Q, plottools:-rotate(P, Pi/3, Pi/4, Pi/8), orientation=[0,0,0]);

The edge-on view of the yellow cube shows exactly the hole you'd need to cut through the blue cube so that a cube of the same size can pass through.

There was previously no convex polyhedron known to not satisfy the Rupert property (which is not to say that all known polyhedra do satisfy it, just that none had been proven not to). But a very recept preprint submitted to the ArXiv https://arxiv.org/abs/2508.18475 constructs a polyhedron the authors call the Noperthedron and they claim the answer to the question "Does it have the Rupert property?" is a resounding "Noperthedron ".

The paper gives a construction of this polyhedron, and it is easily built in Maple

R__z := alpha -> Matrix([[cos(alpha),-sin(alpha),0], [sin(alpha), cos(alpha),0],[0,0,1]]):

C__1 := 1/259375205 * <152024884, 0, 210152163>:
C__2 := 10^(-10)*<6632738028, 6106948881, 3980949609>:
C__3 := 10^(-10)*<8193990033,5298215096,1230614493>:

Cyc__30 := [ seq(seq((-1)^l*R__z(2*Pi*k/15),l=0..1), k=0..14) ]:
# these are the 90 vertices of the Noperthedron
L := map(convert, [seq(M . C__1, M in Cyc__30), seq(M . C__2, M in Cyc__30), seq(M . C__3, M in Cyc__30)], list):

# To actually draw the polyhedron from the vertices, we need to construct the polygons for the faces
H := ComputationalGeometry:-ConvexHull(L):

# A triangulation of the Noperthedron, which is almost okay: all faces except the top and bottom are not triangles
P := [seq((L[x]), x in H)]:

# remove the top and bottom triangles
R1,Q := selectremove(p-> {p[1][3], p[2][3], p[3][3]}={C__1[3]}, P):
R2,Q := selectremove(p-> {p[1][3], p[2][3], p[3][3]}={-C__1[3]}, Q):

# merge the top triangles
T := [map(p->p[1..2], {map(op, R1)[]})[]]:
ht := ComputationalGeometry:-ConvexHull(T):
Tp := [seq([p[], C__1[3]], p in T[ht])]:

# merge the bottom triangles
B := [map(p->p[1..2], {map(op, R2)[]})[]]:
hb := ComputationalGeometry:-ConvexHull(B):
Bp := [seq([p[], -C__1[3]], p in B[hb])]:

# the complete polyhedron:
NOP := [Tp, Q[], Bp]:

plots:-display(map(p->plottools:-polygon(p), NOP), axes=none, size=[1600,1600], scaling=constrained);