WOW! A Most Amazing Answer

It's not so simple to solve for this simple answer. Thanks to all patrons! Special thanks this month to: Richard Ohnemus, Michael Anvari, Shrihari Puranik, Kyle.

WolframAlpha

This is the only channel to feature math topics suggested by people around the world. Support on Patreon to inspire mathematical discovery and make the world a better place:

If you buy from the links below I may receive a commission for sales. This has no effect on the price for you.

Show your support! Get a mug, a t-shirt, and more at Teespring, the official site for Mind Your Decisions merchandise:

My Books
"The Joy of Game Theory" shows how you can use math to out-think your competition. (rated 4.0/5 stars on 39 reviews)

"The Irrationality Illusion: How To Make Smart Decisions And Overcome Bias" is a handbook that explains the many ways we are biased about decision-making and offers techniques to make smart decisions. (rated 3.5/5 stars on 4 reviews)

"Math Puzzles Volume 1" features classic brain teasers and riddles with complete solutions for problems in counting, geometry, probability, and game theory. Volume 1 is rated 4.4/5 stars on 13 reviews.

"Math Puzzles Volume 2" is a sequel book with more great problems. (rated 4.5/5 stars on 6 reviews)

"Math Puzzles Volume 3" is the third in the series. (rated 4/5 stars on 6 reviews)

"40 Paradoxes in Logic, Probability, and Game Theory" contains thought-provoking and counter-intuitive results. (rated 4.4/5 stars on 13 reviews)

"The Best Mental Math Tricks" teaches how you can look like a math genius by solving problems in your head (rated 4.8/5 stars on 5 reviews)

"Multiply Numbers By Drawing Lines" This book is a reference guide for my video that has over 1 million views on a geometric method to multiply numbers. (rated 4.3/5 stars on 6 reviews)

Connect with me

[trystero]

I took one look at it and noticed it was the exact same form as the cubic formula, solved for p and q in the depressed cubic, and then solved for x. Fun fact: you can easily find the two complex roots using the cube roots of unity, which are a great tool for solving cubics :)

[todabsolute]

I spent all day trying to solve this for myself. The solution didn't dissapoint me

[frankcedricfernandez-cabul]

I have to stop pretending I will understand any of your videos and just go on with my life and watch cat videos

[patricksalhany]

Very nice.
Plus this time the solution is well presented.

[jp]

Teacher: what is 2 - 1?

r/iamverysmart: my time to shine is now

[nikosikkurnosik]

In the MAJORITY of algebra problems answe is 1
or 2
or 0

[GrumpyQian]

There is literally a line that says use real valued root instead

[kennethstevenson]

A good way to demonstrate the value of learning to use a mathematical toolbox to solve more complex problems without just typing the problem in and getting the answer. This process reinforces available math knowledge and answers the students question, "What are we learning this for?"(Ex. Binomial Expansions). I am 78 years old, did all of my math learning and computation "Without Computer" and taught Math in Secondary Schools for 35 years.

[wavingbuddy]

Loved it, so satisfying. Manipulation of algebraic expressions is like poetry.

[luminica_]

The subtle and maybe unwilling jab at bluepenredpen at the end.

[StRanGerManY]

Yes! I did figure this one out, thanks for asking :)

[FrPO]

I usually go with this kind of ideas: 1. Guess a perfect cube: (1+√21)^3 =
2. Hence (1+√21)^3/2^3=8+3√21 and you easily take a cubic root of the perfect cube.
3. Same thing with the other term.
4. profit

[Tehom]

The thing that jumps out at me is that their product simplifies nicely.

Let a = (8 + 3 * sqrt(21))^1/3, b = (8 - 3 * sqrt(21))^1/3, then we have:

a^3*b^3 = (8 + 3 * sqrt(21)) * (8 - 3 * sqrt(21)) = 64 - 9*21 = -125

which has a real cube root at -5, so ab = -5. We also have

a^3+b^3 = 16

Let's cube the whole formula, as

x^3 = (a + b)^3 => a^3 + 3 a a b + 3 a b b + b^3

Since the sum of a^3+b^3 is 16, we have:

x^3 = 16 + 3 a a b + 3 a b b = 16 + 3 a b (a+b)

Using ab = -5, we have

x^3 = 16 + 3 * -5 * (a + b)

But x = a+b, so we have:

x^3 = 16 + 3 * -5 * x = 16 - 15 x.

With this, we can read off a solution at x = 1. This checks out. There are
other solutions involving the other cube roots of -125, but they are
likely to be messier.

[samismaiel]

I have found it easier to think of it as

x=16/(x^2+15)
which yields the equation x^3+15x-16=0 with x=1 as a real root.

[haithemharireche]

I really like your videos and all your content
Thank for sharing this with us my friend

[justpaulo]

Beautiful !
Even though cubing seemed at first sight to produce a big mess it also seemed to me to be the only way to tackle this problem.
So I grabbed a pen and paper and started cranking the wheel. Things started to unfold perfectly so I knew I was in the right path.

[RalphDratman]

If there were "no computers allowed" no one would be able to watch this video.

[krishnaats]

Play around with 1+sqrt21 & 1-sqrt21. Or multiply and divide the whole expression by 2. Now use the 2 in the numerator in the cube root and get 64+24sqrt21 and 64-24sqrt21
Since you have already played with 1+sqrt21 and 1-sqrt21, you should know that the above larger values are just cubes of the latter. Now add both and divide the the 2 in the denominator

[sirmarkkevin]

You: WOW! This is most unexpected answer for a simpe algebra problem!

Me: and yet, I can't solve it. 😂

[adb]

I could very easily follow all the steps and understand this simple easy answer.
I could have never come up with these easy steps to get this simple easy answer myself.