x/4 + 8/x = 3 This Algebra Equation is NOT so simple!
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- Опубликовано: 12 май 2024
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![[Find the Differences] #284 Interesting Spot the Differences! 5 questions in total.](http://i.ytimg.com/vi/Z2nMCegzLTg/mqdefault.jpg)
I eliminated the denominators and went straight to the quadratic formula, all in my head and it took max 20 seconds...
Yes I think also this is the easiest way
I did the same thing, clearing the equation of fractions, rearranging terms, getting the quadratic.
Actually it was very simple. I remember solving harder problems in 11th grade.
@@vcambridge-ph.d8030 and i bet it didn’t take 20:13 minutes
Me too, this is childs play.
I try and try to listen to your video but you go off on a tangent too much. Six minutes into the video and you have not begun solving yet. Would love to learn from you but you just talk and talk and talk and ...................................
He just wasted data space
My thoughts exactly. I fast-forwarded it.
X = 8
Do you not know how to drag the video forward?
@@lamper2 I do, but sometimes he will say something pertinent to me but than he goes off again. Not a problem, just won't watch any of this videos anymore. Anywho I'm retired living comfortably, but I love learning.
To add fractions you need a common denominator like you were taught in 7th grade. To do this make the common denominator 4 * x. Then multiply each term by one to get (x/4 * x/x) + (8/x * 4/4) = 3 to get the equation (x^2 + 32) / 4x = 3 . Then multiply both sides by 4x to get x^2 + 32 = 12x and put equation in standard form of x^2 - 12x +32 = 0 . Then factor to (x - 4)*(x - 8) = 0 to get the answer of 4, 8. There is no need to find the lowest common denominator or to multiply the whole equation my some magic LCD of 4*x*1.
Yes, really not that hard. Dod it in my head.
Really there are several ways of approaching this problem which is all relevant.
@@philipclifton7494, Might not be hard to solve for some of us. Problems like these have more meaning to them than just the solution.
It can be hard for people around 7-12 years old. 12+ should solve it quick with use of LCD and solving ax^2+bx+c=0.... (12x=32+x^2) 😀
@@DarVV This clearly a problem for someone that has had algebra 1 in 9th grade.
The answer is x=4 and/or x=8. It's simple by doing it in steps. The first step is subtract 8/x from each side of the equation, so x/4 = 3 - 8/x The next step is to multiply both sides by 4 so x = 4(3 - 8/x) or 12 - 32/x Then multiply each side by a factor of x. X^2 = 12x - 32 Then isolate the equation to one side. X^2-12x+32=0 This factors out to (x - 4)(x-8)=0 or x=4 and/or x=8
Yes it is x=4 and x=8 both work, took me 2 minutes in my head. Step 1 common denominator (4x) x^2/4x +8(4)/4x = 3. Step 2 multiply both sides by 4x, x^2 + 32 = 3(4x) then x^2 + 32 = 12x. Step 3 subtract 12x from both sides x^2 - 12x + 32 = 0. Step 4 factor (x - 4)(x - 8) = 0. Step 5 solve for x in each set x - 4 =0, x - 8 =0. Step 6a add 4 to each side in x - 4 = 0 x - 4 + 4 = 0 + 4, x=4 Step 6d add 8 to both sides in x - 8 = 0, x - 8 + 8 = 0 + 8 x=8 Step 7 test results by substituting results in origional equation. A. 4/4 +8/4 = 3 1 + 2 = 3, 3 = 3,B. 8/4 + 8/8 = 3, 2 + 1 = 3, 3 = 3. Conclusion both 8 and 4 are solutions for x.
To solve the equation x/4 + 8/x = 3, we can first multiply both sides of the equation by 4x to eliminate the fractions.
(x/4 + 8/x) * 4x = 3 * 4x
x^2 + 32 = 12x
Next, rearrange the equation to set it equal to zero:
x^2 - 12x + 32 = 0
Now, we can factor this quadratic equation:
(x - 8)(x - 4) = 0
Setting each factor equal to zero gives us the possible solutions for x:
x - 8 = 0 or x - 4 = 0
x = 8 or x = 4
Therefore, the solutions to the equation x/4 + 8/x = 3 are x = 8 or x = 4.
This is the way I solved it, except just using the quadratic formula once it's in standard form. x = (-b ± sqrt(2ac))/2a, where a=1, b=-12, c=32
4 was the first number that made any sense in my mind and without having much of an understanding of algebra or the other methods used to solve this problem.
I went for 8
Granted, your Marine Corps rants were unnecessary, but they connected with me. I was born in 1958 and spent my high school time reading the Lord of the Rings under my desk through most of my classes. I read through the entire series (Hobbit, Fellowship, Two Towers, Return) each year. So let's say I missed out on most of my Algebra, Geometry and Trigonometry because I did not pay attention. It took me some time to wake up. Five years after I narrowly graduated (1976) I joined the Marines. In 1981 I was able to choose an MOS of Ground Radio Repairman which included Electronics school in 29 Palms, CA. My first course began with mathematics, fundamental to all sciences. In one month, we were all experts at basic mathematics, sort of, at least algebra. I went on to become a Staff Sergeant in the Marines and ran an electronics shop for an artillery battalion. During my advanced courses I learned unit circle, trig functions, and polar / rectangular notations necessary for circuit analysis. After 12 years, I left the Marines and went to college for electrical engineering and learned calculus in college. I ended up with a career in database programming and just retired (Mar 2024). I still love mathematics and often watch videos like this just to keep my knowledge fresh.
Thanks for the video, and Semper Fi, Marine.
Took me about 2 seconds to get 4 as an answer. But to do it on paper, we have x/4 + 8/x = 3. Multiply by 4x to get x^2 + 32 = 12x, or x^2 - 12x + 32 = 0, or (x - 8)(x - 4) = 0, so x = 8 and x = 4 are the answers. This could also be solved using the quadratic formula.
Super simple, I would expect a c student to get this right in less than 20 seconds.
1:) Bow tie method:
(x² + 32) / 4x = 3
2:) get rid of fraction by multiplication:
[(x² + 32) / 4x ] * 4x = 3 * 4x
x² + 32 = 12x
3:) set equation equal to zero and solve via foil
x² -12x + 32 = 0
(x-8)(x-4) = 0
x = 8 and x = 4.
Like I said super simple if you know what you're doing.
If X squared over 4X + 32 over 4X = 3, then cross multiplication would mean that X squared + 32 = 3(4X), simplified to X squared + 32 = 12X. Then X squared minus X + 32 = 0. It then becomes (X-4)(X-8)=0. X-4=0, then X = 4. X - 8 = 0, then X = 8.
This was a good one. I labored for a long time before finally realizing that I had to get that x out of the denominator. So here comes the quadratic equation, and it was simple to solve.
Could you just have looked at it, and it is obvious that if you sub in x=4, both sides are gonna be the same?
@@marscience7819 Yeah, I kind of did a facepalm when I got the answers. Sure, one number plus another number equals 3. The first choice would be that one fraction is 1 and the other is 2. That would have been so easy. But I knew this was an algebra problem, so I tried to use algebra to solve it.
@@marscience7819and you missed the other value. there's actually two values for x.
In Hong Kong, some primary students can complete this simple calculation. I used to educate Olympic math and eye level. Some primary students can handle this easily
Simplify the equation to the standadrd quadratic x^2 -12x+32=0; factorising you get (x-8)(x-4)=0. Hence X=8 or 4.
"Simplify the equation"
Increasing the order (from x to x^2) is not simplification. your mileage appears to vary.
I eyeballed it at 4. When you said 2 answers I paused and double checked that 8 would also work.
I am happy to inform you that your lessons have produced solid results.
6 months ago I would have given up on this one and now I can solve it using algebra.
Now I'll watch the video to see if my method was the right one but I'm confident it was. :)
Thanks!
Can't you just look at it, and it is 100% obvious that if you sub in x=4, it will work?
@@marscience7819 Yeap! But if you want to deal with tougher equations you need to learn algebra. In fact, I did not even think about plugging in the number 4 when I saw this problem.
In other words I went from thinking like you did- totally fine- to thinking algebra first.
So, I stand corrected in the sense that I would not have given up on the problem but I don't think the same way since I worked on my algebra skills which are still a work in progess BTW. ;)
Try this one and see how you are doing without algebra: x/89 + 143/x = 542
Just a hint: number 4 won't work. ;)
I solved this intuitively, and now that I’ve watched the video I’m sure I couldn’t solve it. I give up.
X = 8 or x = 4
I think I've solved so many of these this lifetime I could see the answers in front of me. The first obvious one was 4. Because it was a quadratic equation and I instinctively knew the answers were whole numbers figuring out 8 took another 35 seconds (I was timing). Not exactly procedure I'll admit.
Yep and it was super simple
When solving equations you move terms from one side of the equals sign to the other and change the sign. Positive terms become negative, division becomes multiplication etc it is much more efficient.
So solve for X and show your work.
It's simple enough, that I don't need to create the full quadratic equation. The mean of the roots is 3 (constant term) * 4 (recibrocal of linear term) / 2 = 12/2 = 6. Obviously x = 4 and x = 8 result in 1+2 or 2+1, both are 3. Or you calculate 8 (1/x term) * 4 (see above) = 32 = x_1 * x_2. Or you take the long way with x^2 - 12 x + 32 = 0 => x = -(-12)/2 +/- sqrt(36 - 32) = 6 +/- 2.
Multiply everything by 4x...
4x² - 12x + 32 = 0
(x - 4)(x - 8) = 0
x = 4, x = 8
This is always where I got in trouble in school. They always wanted to show the work. They wanted to see how we solved it. That takes too long. I just look at the problem and my brain figures out the answer. This happens almost instantly
I think I knew this sinds 1962: solution x =4 and x=8; by writing first part as the sum with the common denominator 4x and multiply by 4x you get the quadratic equation:ax^2+bx+c= x^2-12x+32=0; D=b^2-4ac; x=(-b±√D)/2a (D is the Discriminant)
you did it the hard way with the -b part, but yes technically that is correct. Now, if the test giver had requested foil or factoring, then no this would be wrong.
Keep the videos to 5 minutes maximum. I like learning but 16-18 minutes bores me so I have stopped watching/learning.
Multiply by 4x to eliminate denominators.
You get a quadratic; apply the quadratic formula.
Or, even better, in this case the quadratic easily factors into a product of linear factors
(x-4)(x-8).
The key to solving is to get all of the terms under a common denominator. The result is a quadratic equation in the numerator, which is easily factored into the two solutions.
Exactly.
x/4 + 8/× = 3 = ?
cruzando as duas frações, vem:
x elv 2 + 4 X 8 / 4x = 3
x elv 2 + 32 = 12x
x elv 2 - 12x + 32 = 0
que é uma equação do 2° grau
a = x elv 2 =
b = -12
c = 32
Dois números cuja soma seja - 12 e o
produto seja 32.
Ax + Sx + P =0
S = -4 + (-8 ) = - 12
P = -4 X (-8) = 32
Entoa as raizes da equação terão os sinais trocados:
r1 = 4 e r2 = 8.
(x/4)+(8/x)=3
therefore x not= 0
x × {(x/4)+(8/x)=3}
x^2/4+8=3x
4×{x^2/4+8=3x}
x^2+32=12x
x^2-12x+32=0
note => 2 solutions for x but not 0
the quadratic equation:
[-b +/-sqrt(b^2-4ac)]/2a
for
x^2-3x+8=0
a=1
b=-12
c=32
so
[-b +/-sqrt(b^2-4ac)]/2a
[12+/-sqrt(144-128)]/2
[12+/-sqrt(16)]/2
[12+/-4]/2
sol.1 x=4
sol.2 x=8
verify
(x/4)+(8/x)=3
sol.1 x=4
4/4 +8/4=?3
1+2=❤3✔️
sol.2 x=8
8/4+8/8=?3
2+1=❤3✔️
Gosh u r quick!
Beat me to the answer set:-D
X =4 or 8. Multiply through by 4x and you'll have a quadratic so make it equal zero and use the quadratic formula to solve
I multiplied by 4x at the first step for convenience, brought all the terms to one side and factored. Took 3 or 4 lines to the solution.
Guess and check is quickest in this case. I need 2 numbers that add up to 3. Hmmm 1+2=3. Looks like 4 works in that case. Algebra is not always needed, sometimes a little number sense can work.
X=4 & 8.
Did it in my head & within a few minutes.
got it 4 & 8 solved by sight easy calculation had to be 2 + 1 and 1 + 2 thanks for the fun
yea it was real easy
At 08:45 you have obviously decided to make the process as complicated as possible. I was able to solve this by simply looking at it for a few seconds.
Shalom to everyone
Does the teacher John have work sheets. About 100 drrills per sheets ?
Thanks.
It is simple, if we state, that x is ot 0. In this case, we can multipl the equation withh 4x and get thhe quadratic equation.
x^2-12x+32=0
which leads to the 2 solutions:
x=6+-sqrt(4)
x=4 or x=8
We can state, thhat x is not 0, because otherwise the term 8/x is undefined.
I did not even think about using lcd. I just multiplied both sides by x and right there you have a quadratic equation… And, while factoring this out is nice, I am lazy and would just plug it into the formula😎
That's how I did it. The quadratic formula is burned into my brain
4/4 + 8/4 = 3
8/4 + 8/4 =3
X = 4 and 8
It was to obvious, but I don't remember how to find X if the solution had been more complicated. Expect Ill find out in the rest of the video.
After 50+ years of taking algebra, I got this with no problem. Easy peasy.
Thank you John, I appreciate your content.
I had 4 and 8 on sight. I am a 61 year old British electronics engineer who mainly uses standard equations, calculus and FFT.
It all results in an equation x² - 12x + 32 = 0 so (x-4).(x-8) = 0 and x = 4 or x = 8
Great content. How about just 1 tangent per video?
You're a great teacher. Keep doing what you're doing.
Use LCD to get the variable out of the denominators. Good Stuff.
Great little problem, especially making sure that people check the answers actually work. Thanks for creating and sharing. I love your channel.
However, I feel that it would be easier for people to understand if, rather than distributing by 4x, the fractions were both converted to 4x denominators then everything multiplied by 4x to produce the equation that becomes the quadratic. As usual with a lot of problems, always more than one way to solve and it’s about finding the method that works for the student.
What i don't get is the reason "Why" you put -12x in front of +32 =0. I cant understand the problem unless that's explained.
Intuitively 2+1=3, 1+2=3, but I couldn’t apply my procedure to other problems hence my weakness. I need to learn procedural steps that will apply to other problems.
Resolves to a quadratic x^2 - 12x + 32 = 0, I got the root numerically = 4 or 8
I have to agree with Shelly. When I do one of your problems--I just skip ahead. In the words of my daughter when asked why she didn't want me to help with her math, she told her mother, "Mom, I don't want to know that much."
Excellent video!
Can you get to the f323cking point?
Solved by going to quad form with some machination and using the quad formula. Year out of practice.
I figured this out in my head. What numbers added together equal 3? 1 and 2. So the fractions have to equal either 1 or 2. So X can be either 4 or 8. Too bad that it takes such
roundabout, convoluted methodology in order to prove this.
my TI-84+ CE cant solve this. It says zero/error.
See sir this is a mathematics to be calculated by Brainly,tricky,bodamus,pedamus.i don't think so general way of solving for solution is not necessary here.thank q for understanding me.
Took one look to ralize straight away just a quadratic equation -multiply through by 4x to make it look normal.
I didn'y bothering spending 20 minutes to watch the video.
Just needed 15 seconds to do this in my head. This could have been explained on the board in two minutes.
X/4 + 8/X = 3
Muktiply both sides by 4X
Gives
X² + 32 = 12X
X² - 12X + 32 = 0
(X - 8) (X - 4) = 0
X = 8 or X = 4
By inspection, x=8. Took about 30 secs. On further thought, x could also be 4.
I instantly divided into c (each x turning into 1.
1x8=8
1x4=4
=8/4
=2
1+2=3
I love these math puzzles. Good for the noggin, which is ancient.
Mental math 101. Look, think, solved it! Something + something = 3. Answer: 4 😊
Thank you.
Ok, if you want to go into quadratic equations. But simple logic gets y 13:04 13:04 13:04 y 13:04 y 13:04 ou there in seconds: 3 is a positive integer. Therefore no fractions possible. X therefore multiples of 4. Minimum 4, maximum 8. Try both. Voilà!
Combine the fractions with a common denominator, multiply both sides by the common denominator, rearrange into a quadratic and factor.
All of that I was able to do in about 29 seconds. It should not take 20 minutes and 14 seconds to explain that.
I solved this in 10 seconds. Easy. Multiply both sides by 4x, the lowest common denominator. x^2 + 32 = 12x. x^2 - 12x + 32 = 0. (x - 4)(x - 8) = 0. x = 4 or 8
Thank you Sir, for you're service too. Semper Fi! Sir I just added the 4 + 8 (adjacent diagonal numbers) = 12 / (4 lcd) = 3, x = 4 , easy peezy lemon squeezy right? Amen?
you missed an answer there's actually 2 values for x
I eyeballed this "difficult" problem and had a working value for x in 20 seconds. When the sum total is a low integer like 3, anyone can iterate the likely possibilities and solve for x without any of the complicated algebra you claim is "required." I did!
Thank you for your service 🙏 ❤️ 🤍💙
Use 4 instead of 3 to make the problem non trivial. Then explain it in 2 minutes instead of 20. LHS = (x^2+32)/4x so x^2+32 = 16x or x^2 -16x + 32 =0 ➡️ (2.34, 13.66) . CHECK ( sum of roots = 16). Students dont appreciate being shown how to solve a problem that can be done by inspection.
"...don't remember much of anything other than getting yelled at..." 😆😂
me: USMC 1963-69
Semper Fi, Brother!
I had it with the bow tie method in the begining with 96/32.
Loved this… Thank You 🙏🏻
Thank you
No need for complicated algebra. Use cross reduction and you get 1+2=3. The X’s cancel each other out, 4 and 8 a divisible by 4 leaving you with 1 over 1 and 2 over 1.
"Use cross reduction"
That only works *multiplying* fractions. This is addition. Anyway, what is wanted IS the X and you eliminated it completely.
@@thomasmaughan4798 so, I simply solved the equation. So, you’re right I didn’t solve for X. However, cross reduction is a form of simplifying fractions to their lowest form and does work for addition and subtraction of fractions.
@@potterspride1 Well, whatever. I trust you do not build bridges.
@@thomasmaughan4798 no, I don’t build bridges. However, if I did, I wouldn’t be using algebra to build it. I would use exact numbers, and if a number wasn’t known, I would go out and measure it again myself.
@@potterspride1 "no, I don’t build bridges."
That's a relief!
Maxima quickly finds that only 2 answers exist: x=4 and x=8
Lol! Good review. I guess -20 won't quite fly! How'd I do that anyway?Thank God I don't work for Boeing!
Thanks for sharing 🙂
That's so easy. You can see it has to be 4 and 8 just by looking at it!
Greetings. Yes, we can. The answers are X=4, 8. The first thing that has to be done is to get rid of the fractions by multiplying throughout by 4X to get X^2-12X+32=0. Thereafter, we will factorize the equation to get
(X-4)(X-8)=0. That is (X-4)=0, (X-8)=0 and X=4, X=8.
I looked at the problem for about 15 seconds and easily deduced x=8 AND x=4
How is this difficult?
Well, I did it by taking care of the fractions first and then used the cross product to solve the problem.
At the end I came up with the right equation anyway i.e. x Squared + 32 = 12x
Then I substracted 12x from both sides, factored and came up with (x -4) (x - 8) = 0 and voilà!
But I could have done it by multiplying the whole equation by 4x as you did.
When I see fractions and an (=) sign I always think cross product. ;)
-> x^2-12x+32=0 , x^2-4x-8x+32=0 , (x-4)(x-8)=0 , x= 4 , 8 ,
HS Algebra in 1960s and it took me all of 2 minutes to do this in my head.
Ah, but you need to show your working
@@kbstabs5982 x/4+8/x=3 ->multiply by lcd 4x -> x sq. +32 = 12x -> x sq -12x +32 = 0 -> factor (x-4)(x-8) = 0 -> x=4 and x=8 Took longer to type it out than to think it through.
Yes it is simple. Just find a common denominator (4x) and rewrite the equation in terms of the numerators, re-arrange the terms, and complete the square. x = 8 or 4.
super simple don't let the variable in the denominator scare you off. Like Sheldon Cooper, always wear a BOW TIE.
I knew right away that it needs to be 1+2=3, so the real problem is to write it down I need a 1 and a 2, 8 is 2 times bigger than 4 so its kinda given already that one solution is 4 another is 8 because 2+ 1 is also 3. So I would solve it by X/4 = 1 == 8/X = 2 so X/4 = (8/X)/2... (X/4) *4*X = (4/X) * 4 *X ... X*X = 4*4... X = 4.
Like anyone with basic understanding will know that 1+ 2 is 3 and they will also notice that 8 is 2 times bigger than 4 so there you have the 2 so you now only need a 1 and a number divided by itself is 1. So I would have written X1= 4 X2 =8 and got 0 points because you should write a half page of BS to prove it even if you never asked for all possible X's and its sort of given from start... Our brains love shortcuts so forcing it to do stpd work... once again do you want to fck with em or...
My point is probably that make a problem that has to be solved in the correct way to even see the answer, don't show the answer and force em to prove how to get to it. Like when you get a puzzle game, do you really want to know the solution before you solve it. Our minds work in different ways, some love easy and love to do it the way it is in the book(at least the bookworms that has the natural talent to be able to learn by reading). Then suddenly we are allowed to take shortcuts with the calculator even if its not using your brain to solve it... simple calculations like 25*25 is allowed to skip but x/1 + 2/X = 3 has to be proven even if you already see the answer. I know that you need it for later but why allow shortcuts like calculators at all and why even present a equation that you solve within 10 secs just by looking at it.
I actually got there but didn't trust my answer as I couldn't remember the reasoning.
I didn't do any algebra for this. I just looked at it and knew that 1+2 and 2+1 equal three. So the answers have to be 4 and 8.
I did that one in my head in 8 seconds.
Since the answer was a whole number with no fraction and there are 2 even numbers in the equation , then x = even number. I tried 2 (nope) then 4(yep).
If you wish to skip over the yakkity-yak go to 3:40.
x = 8 or x = 4. Since this is a Quadratic equation, it should have two solutions.
This is about as simple as any factorization gets. I'm 68 and didn't find this at all difficult. Why the title that says it is?
4 quarters equals a dollar 8 quarters equals 2 dollars together they equal 3 dollars way easier
Nonsense.
This is one of those annoying equations that have two answers.
I vaguely remember it from school - I think it’s 4 or 8, without doing the algebra, cause I can’t remember how to do it!!!😂
x=8 solved in 3 seconds. Of cause it can technically be 4 as well, so what is the point of having a equation where you can not know x from it?
How is it not so simple? It's an extremely standard exercise.
You should always use the quadratic equation. When I discovered the quadratic equation I was mad that math teachers wasted my time with factoring.
x/4 + 8/x = 3
x²/4x + 32/4x = 3
x² + 32 = 12x
x² − 12x + 32 = 0
(x − 4) (x − 8) = 0
x₁ = 4 ∨ x₂ = 8
𝕃 = {4, 8}
Said to self... Looks like 8. Yup. It's 8. Forgot that there could be more than one answer.
Bro why go through the song and dance?
3= x/4+8/x
X=4 1+2=3 ?