solving an exponential triangle problem

Here we have an exponential triangle problem, i.e. find x so that e^x, e^(2x), and e^(3x) form the sides of a right triangle. We will need to use the rules of exponents and even the quadratic formula for this one. This fun math tutorial will provide a challenge for your algebra 2 class. Check out the logarithmic triangle problem • Solving a triple natur...
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"Just Algebra" (by blackpenredpen) is dedicated to helping middle school, high school, and community college students who need to learn algebra. Topics include how to solve various equations (linear equations, quadratic equations, square root equations, rational equations, exponential equations, logarithmic equations, and more), factoring techniques, word problems, functions, graphs, Pythagorean Theorem, and more. We will also cover standardized test problems such as the SAT. Feel free to leave your questions in the comment!
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#justalgebra

[@oenrn]

For anyone wondering, the sides are √φ, φ, and φ√φ.

[@Justinr-12r]

tf are those symbols

[@metehan9185]

@@Justinr-12r this is phi. Phi is the angle of the z axis that is in the case used for the golden ratio phi = sqrt5+1/2

[@matt-fitzpatrick]

That is so cool that phi does that

[@HappyGardenOfLife]

@@metehan9185 Shouldn't that be sqrt5/2 + 1/2?

[@furezuu]

@@HappyGardenOfLife Probably they forgot to put parethesis there

[@Dalton1294]

Another way to write the answer is (ln(1+√5)-ln2)/2

[@aerodataspace4244]

I love math, specifically quadratics

[@StanLiao]

"My name has two e's." I count 3 e's in "blackpenredpen" 😁.

[@dwagonyt]

Love the video, quick suggestion: Could you solve a right triangle with sides: sin(x), cos(x), and tan(x)? Let me know!

[@meurdesoifphilippe5405]

Two cases are possible : if tan is the hypotenuse then tan^2 = sin^2+cos^2=1, and sine and cosine >0 so x=π/4 +2kπ.

[@dwagonyt]

@@meurdesoifphilippe5405 hey thanks so much for that; I’m not the most mathematically fluent but when you say tan^2 I’m assuming you mean tan^2(x) and same with the other functions, and for the solution for x, what is k? Thanks!

[@dwagonyt]

Still though, I would love to see a video on that !

[@emm3rsk]

@@dwagonyt k is any integer you want

[@degenerationz9284]

@@dwagonyt k∈Z

[@ednk5054]

"No complex triangles here" So funny dude, got my like))

[@pad-dydrummer5929]

I will never be able to use two markers so conveniently haha! 😀😀🔥🔥 Crazy one as usual!

[@shruggzdastr8-facedclown]

I've seen double-ended markers before (one color at one end and a different one at the other)

[@saharhaimyaccov4977]

Do it for sin(x) - sin(2x) - sin(3x)

[@SuperYoonHo]

Wow this fits perfect with my morning(all-day) math i homeschool (don't go to school) with your videos and those help really a lot. Thanks mate and hope (wish) you would get many_subscribers!(factorial)

[@UnPackedWolf]

Honestly it feels so good to be able to solve this

[@kaistrandskov]

This just blew my mind. Someday I'd love to see the connection between phi, e, pi, and i. But for now, my mind is blown for the day. thank you!

[@samwalko]

I remember seeing a problem with a very similar answer quite recently on some other math channel. Edit: nope actually, oops. It was this channel a couple weeks ago, with e^x + e^2x = e^3x. Solutions are just double what they are here.

[@bobthegoat17]

With school coming up I gotta watch more of these and hope I'll understand one some day

[@williamhogrider4136]

That's very cool 🍺🍺🍻.

[@pimpaovieira]

Professor, can you make a class, Explaing how to solve cubic equations? Sorry, I dont speak inglish very weel... I am brazilian!!

[@ladripper47874]

2:50 why does he not proceed with making it 1+e^2x-e^4x=0 and then rearranging to -(e^2x)^2+e^2x+1=0 and continue from there? I'm not terrible great at math, but I feel like I have an error somewhere

[@user-wu8yq1rb9t]

I'm big fan of *i* . Nice ..... Thank you Teacher ❤️

[@sphakamisozondi]

I'm a big fan of π

[@user-wu8yq1rb9t]

@UCesJUCFKPvBq6Mnr821C3dA I like *π* too and also *e* but complex numbers are another thing to me .... *It's shrouded in mystery*

[@NoNameAtAll2]

"my name has 2 'e's" blackpEnrEdpEn - that's 3 :)

[@hkayakh]

Two e’s in your name? But ‘just algebra’ Has one!

[@Serghey_83]

e^(2x) =φ = [1 + sqrt(5)] / 2 a = exp(x) = φ^(1/2) b = exp(2x) = φ c = exp(3x) = φ^(3/2)

[@hafizusamabhutta]

What's your name?

[@lassikokkonen5618]

Bruh you for got negative infinity

[@dorol6375]

I got x=-goldenratio÷2

[@nimmira]

Just realized this... Canon EOS 7D's sensor's crop factor is 1.613, just 0.005 shy away from φ

[@KingGisInDaHouse]

I was going to do something similar pun intended. I knew x wasn’t going to change and that if this was a real triangle x is greater than zero so for the purposes of solving for x you can just divide out the e^x on every side of the triangle. Now it’s a 1,e^x,e^2x 1^2+e^x*e^x=e^(2x)*e^(2x) 1+e^(2x)=e^(4x) U=e^2x 1+u=u^2 I skipped the work here because I knew that phi squared is phi plus one already I’m assuming you worked this out for your audience. u=phi or phiBar u=e^(2x) ln u=2x x=ln(u)/2 x=ln(phi)/2 x=ln(phiBar)… no bueno phi bar negative ln(neg) no existy in x€R

[@truejeffanderson]

Kepler's triangle

[@debdami]

the hypothesis that e^x e^3x ?

[@harshrajsinhsarvaiya3024]

I am watching this video at 2 am. Lots of love from India 🇮🇳❤️

[@Selim_Osmanli]

Thank you very much goodsir all respect and aprishiation, love your videos keep them up 💥🥳

[@andreasobama9487]

what in the hell

[@dean532]

Yea, where do you get your T-Shirts from? 😅

[@alejrandom6592]

Ahhh yes, ln(φ) also known as arcsinh(1/2)

[@klementhajrullaj1222]

And if sinx, sin2x, sin3x, how is the solution?! 😀😉

[@del66404]

👏👏👏👏👏👏

[@OptimusPhillip]

x=ln(φ)/2

[@hans429]

Wen ever i see the prof, it means pain to my brain 🤣🤣

[@giuseppemalaguti435]

x=ln@/2

[@mryjunior9391]

Hi, isn't the solution at the end meant to be x = ln(sqrt(phi))? Because e^2x is the same as (e^x)^2 so you have to square root first then natural log. Just curious, great video otherwise Edit: just realised ln(sqrt(phi)) can be written as ln(phi)/2 (because of logarithm power rule) sorry about that

[@lukasstanik1107]

Unless I am missing something, isn't it what he got at the end? Check 4:56

[@mryjunior9391]

@@lukasstanik1107 ye you're right I forgot to edit the reply ln(sqrt(phi)) can be written as ln(phi)/2 my bad

[@mel_ha22]

OMG, I didn't understan the end part because I have never seen the In thing 😫😫

[@Grizzly01]

ln is the natural logarithm aka logarithm base e

[@ItsPungpond98]

Learn Chinese with bprp

[@ccbgaming6994]

Hey, I was wondering if you or any of the smart people in the comment sections can help with a math problem. 5^x + 6^x + 7^x = 8^y + 9^y + 10^y There should be an infinite number of solutions that can be found by an equation (y = …) that can be made from this one. The question is what is the equation. There should only x’s on the other side

[@perfectsinister2856]

in which set of numbers is x and y defined?

[@ccbgaming6994]

Real

[@Pratyush_jha590]

@@perfectsinister2856 real

[@metehan9185]

2:36 why cant we do 1+e^2x-e^4x = 0 ?

[@dwagonyt]

He is trying to break up the exponent of e^(4x) into e^(2x)^2, but of course you could just bring it to the other side and say: -e^(2x)^2 + e^2x + 1 = 0 and use the quadratic, however a negative a value doesn’t always look the neatest, that’s all

[@metehan9185]

@@dwagonyt thanks

[@dwagonyt]

@@metehan9185 no problem

[@zidanihamid7485]

Beautiful

[@p12psicop]

1.27-12.73-45.42

[@pranavamali05]

thnku

[@ghiurcarobertnicolas5957]

Wait a bit…. e^6x/e^2x is e^3x not e^4x , is it not?

[@adibmustafa914]

6-2 is 4

[@IMAN_IIT_PATNA]

Lol 🙂

[@robertveith6383]

No, you wrote it wrong. e^(6*x)/e^(2x) = You must have grouping symbols.

[@oenrn]

When dividing powers with the same base, you subtract the exponents.

[@adityakumargupta9730]

There are infinitely many values of x, for which this "exponential triangle" exists. x = (1/2) ln[(1 + Sqrt[5])/2] + (2 k Pi) i - where k is any integer. All the values above make (e^x) = √φ.

[@TrendingShortsA1]

Sorry, but there will be - e²x/e²x=1, e⁴x/e²x=e²x e⁶x/e²x =e³x

[@datarioplays]

When dividing, you SUBTRACT exponents. For example (2^8)/(2^3)=2^(8-3)=2^5