x/4 + 8/x = 3 This Algebra Equation is NOT so simple!
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- Опубликовано: 6 фев 2025
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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.
Hi, I'm 75 but still wamt to learn. I was always intimidated by math so I'm trying to conquer my fear. Thank you. Judy from Birmingham
I'm 5 years younger and when I was in school, the minute girls hit teen years, math was for boys, not girls for the most part. If you liked math you were not popular.
Keep the videos to 5 minutes maximum. I like learning but 16-18 minutes bores me so I have stopped watching/learning.
Only the ignorant are bored!
@@7dimensions-jebname calling is also a common denominator with ignorance…
Followed quite a few of his but i give up screaming here. Too much nervous energy and chat, and oh by the way …..not easy to take notes when he’s going off on digressions and sales pitches in between youtube ads. Too frenetic and nervous chaos.🤯 I’m sure a ton of content you have, but kindly remember kiss. Stop telling us about that and get on with showing us instead of interrupting yourself every 5seconds. I’m going back to the books after this.
Someone who hates math but needs to learn it isn't gonna wanna watch A 20minute video on math@@7dimensions-jeb
@ Thank you I didn’t know that.
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.
You expand 30 second problems MAX into a 20 minute video all the time
Indeed, being succinct is an ART ;-)
4,8
He is aiming towards early high school. I have a degree in math. He adds humor to math. There are other content creators that you may look into.
He's solving to teach, not just to solve. He is also promoting (selling) his courses by demonstrating his thoroughness. Definitely not a hard problem for those who are skilled in algebra. Not so easy for a 9th grader in their 1st algebra class.
That's what you call good teaching. It may be a 30 second problem for someone who knows what they're doing, but he breaks it down and covers all of the bases so even a moron like me can understand it.
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.
you lost me at step 3, how did you end up with 0?
@@samanthadry9214 in order to factor the equation must = 0 this puts everything on one side. If it wont factor the easy way there is another method.
But why Does X have 2 different answers? Solve the mystery of the Universe for me.
I had 2 years of math back in the 50s !. I’m so glad the way I learned algebra was not as complicated. I resolved this problem in seconds.
I had to retake an algebra course in my late thirties since it was a pre-requisite to a university certificate I was taking. I am now 67 and I solved the problem like you in seconds. He had me doubting myself since he taking so long to explain it.
Interruptions of explanations to promote other programs is unhelpful.
I eyeballed it at 4. When you said 2 answers I paused and double checked that 8 would also work.
Same 😊 I was always in advanced math in school and I quickly realized that "I still have it". I also realized that I still have the same problem. I was able to plug in a number by eyeballing it and the number was right but I would not be able to show my work. And I always got deductions for not properly showing my work 😅
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 I did it that way mentally
@davidseed2939 I remember in school the instructor wanted students to show their work to explain how they arrived at the answer, so the solution was just as important if not moreso than the correct answer(s).
x/4 + 8/x = 3
x +32/x -12 = 0 (times 4)
x² -12x +32 = 0 (times x)
Then just apply quadratic formula
x/4 = y, y + 2/y = 3.
y**2 - 3y +2 = 0.
y' = 1, then x=4.
y" = 2, then x=8.
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'm using my RUclips Premium subscription to learn freestyle swimming, math, Hungarian, Greek, Mandarin Chinese, and other things. But being 72 years old I want each episode to be as succinct as possible without unnecessary talk
I don't even know why... but, I found this challenge so satisfying... I actually remembered... I'm 1966.
Thank you for the confidence boost. Makes me feel like I can help my grandkids.
I was good at math in 2000, 24 years later my daughter is struggling with this. I need the refreshers on all the rules, I like that you include the methods and explain them in each video. Very helpful so I can help her
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!
x² + 32 = 12x, x² - 12x + 32 = 0, (x - 4)(x - 8) = 0, x = {4, 8}.
I plugged each answer into the original equation and they both worked.
You drag it out too long.
He babble too much
@@eugenewilliams8440,
he asks you and me to understand that talking is much more amusing than algebra .
Two of the people who replied to the above comment need refresher courses in English.
It would give Young Sheldon pause.
taught math to most of my engineering class for 4 years. Went to work and became a digital designer who uses almost no math. I am 77 and love relearning, thank you!
I died of old age watching this video 😆
You are bringing back lots of teaching I have had. My first algebra class was 1965!!! Thank you!!
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
Thank you for your service. I was denied the honor in 1976 due to being an Epileptic, but so grateful to those that can and do
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.
Thank you for taking the time to explain! I am enjoying the learning process!
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.
what software and devices do you use or your videos?
x/4 + 8/x = 3 -> x^2/4x + 32/4x - 3 = 0 -> ((x/2+32)/4x) - 3 = 0 -> x^2-12x+32 (Quadratic equation)
Given this quadratic equation we can solve for solutions to x using the quadratic formula: (-b+-sqrt(d))/2a where d=b^2-4ac is the discriminant to the quadratic equation:
x = (-b+-sqrt(b^2-4ac))/2a = (-(-12)+-sqrt((-12)^2-4*1*32))/2*1 = (12+-sqrt(16))/2 = 6+-2
The discriminant (16) is greater than 0 so we get 2 solutions:
x1=6+2=8
x2=6-2=4
Intuitively the solution could also have been found by simply looking at the equation x/4 + 8/x = 3 and realizing that probably the solutions are 4 and 8 by substituting into x and doing a bit of simple math: 4/4+8/4=3 as well as 8/4+8/8=3. With more complex equations and/or unintuitive values of x the best method is to use formula for the discriminant and the quadratic formula.
I followed and loved you solved the algebra!!!!
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.
I did that too, but stopped at 4.
A little more persistence could have given "What about 8?", since 8/8 is 1, and 8/4 is 2, and 2+1=3. Once you see it explained, it seems too simple.
x/4 + 8/x
This particular equation can also be factored into 2 Binomials sq. by each other
(x-4)*(x-8)=0 and {x = 4; x = 8}
x^2 - 12x + 32 = 0
x/4 * 4x/1 + 8/x * 4x/1 = 3 * 4x
fours cancel out, and leaves you with x^2; 8/x * 4x/1... the x's cross-cancel and leaves you with 8 * 4 = 32;
Then, whatever you do to one side always do it on the other side of the "=".
that gives you 12x
quadratic equation: Ax^2 + Bx + C = 0 (in standard form)
or (-)b±√b^2 - 4*a*c/2*a
You should always use the quadratic equation. When I discovered the quadratic equation I was mad that math teachers wasted my time with factoring.
Agreed! Factoring is trial and error guess-work - just hoping that known factors of 32 just happen to add up to a nice round number like 12. 40 years of working with computers tells me that a simple function works best: x = solve_quadratic(1,-12,32) - sorted! 🤓
I really like your video. You are very thorough and try to point out the complexities by referring to many points of solving the equations. Yes it may seem tedious but it helps me. I still have some trouble knowing when I need to look for lowest common multiplier versus lowest common denominator. I think it takes a lot of patience and determination to get good at this stuff. I am a biologist and though I've had some training in math, I was always kind of slow, but usually I could get through the courses with a good grade. Now I am retired and I find it a fun game to redirect my attention to algebra and other math, so I hope you and I will keep enjoying the challenges. Keep up the good work, and best wishes!
You explain to much which deviate from yr target.we don't have too much time to explain the problem.Pse shorten the explanation.
Please remember there are others here who are not as knowledgeable as others. Have patience with the slower ones, ok. This is a how teaching goes.
1)- Thank you for taking the time, but you took us around the world and brought us back to the beginning.
2) - Algebra is not as easy as you have tried to teach your pupils or students.
3) recommend you to teach your skills in an easy meeting!!!!.
(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
@@tomtke7351 Fast and accurate
Relatively simple proof of quadratic formula:
ax² + bx + c = 0
(ax² + bx + c) /a = 0 /a (a≠0) (simplify x²)
x² + bx/a + c/a = 0
x² + bx/a = -c/a (isolate x terms on left)
x² + x•b/a + b²/4a² = -c/a + b²/4a² (completing the square on left by taking 1/2 of b/a = b/2a and squaring it, and adding to both sides)
x² + bx/a + b²/4a² = -4ac/4a² + b²/4a² (combining terms on right)
x + b/2a = ±√(b² -4ac) / √4a² (taking square root of both sides)
x + b/2a = ±√(b² -4ac) / 2a
x = -b ±√(b²-4ac) / 2a
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
I took algebra in my first year in high school as a freshman. First we find the common denominator, which is 4x. Then, we multiply the 4x both sides which gives us x^2 + 32 = 12x. Then we make the equation to equal 0 by subtracting 12x both sides, which is gives us a quadratic equation of x^2 - 12x + 32 = 0. Thirdly, we factor the trinomial into two binomials using the FOIL method which is (x-4)(x-8) = 0. Finally we make each equation to equal 0, which is x-4=0 and/or x-8 = 0 and then solve for x. We get x = 4, x = 8.
He’s TEACHING slow enough for children & teenagers to grasp. I think many of you look at this to test yourself & memory which is not what it’s intended for. Unfortunately, many folks being paid as teachers in public school are not teaching to educate the kids or help them comprehend the information. I feel he has a real grasp on teaching.
Unlike your math teacher, you can pause this. It's not only the speed it is the relentless discursive rambling!
This problem is EXTREMELY SIMPLE!! The result MUST equal 3, and "X" must be the same number. In this case "X" = 4. So, 4/4 = 1 + 8/4 = 2 Guess what? 1 + 2 = 3 Kindergarten wasn't wasted on me.
😂😂😂
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.
Been many years for me since school. Thank you.
X=4 & 8.
Did it in my head & within a few minutes.
This one is easy by inspection. 1+2=3, 2+1=3
1) x/4 + 8/x = 3
2) get "rid of the denominator 4 so multiply BOTH sides of the equation by 4. This leads to: x + 4*(8/x) = 4*3
3) simplify it a bit: x + 32/x = 12
4) Multiply BOTH sides of the equation by x: x*x + 32 = 12 x
5) Rearrange and simplify: x^2 -12 x + 32 = 0
6) looks like special product (in this case fairly simple to me, I suck at factorizing) (x-4)*(x-8)=0
7) (x-4)=0 or (x-8)=0 so x=4 or x = 8
8) Cheers and have a nice day.
Ya'll stop complaining about him being precise in his teaching. The classical style demands he not leave out a step
There seem to be a lot more 'rules' and 'special methods' that have to be remembered and applied in maths these days. I learned algebra in the mid '60's using very basic and fundamental rules. Which I still remember, despite my geriatric decrepitude.
Rule #1.
As long as you do the same thing to both sides of an equation you are good to go.
In this example, to get rid of the fractions multiply both sides by 4, THEN multiply both sides by x. You then rearrange to get a simple quadratic
x squared - 12x + 32 = 0
It is much easier to follow if you do each of the previous steps separately.
Then, either factorise the quadratic:
-4 + -8 = -12 and -4 x -8 = 32 so the quadratic becomes (x-4)(x-8)=0
Or use the quadratic formula to get x = 4 and x = 8 directly, but that I feel is cheating by doing things by rote, but it is one of the things I did learn.
By inspection, it can be seen that both these solutions give the correct answer to the problem when you plug these numbers in for x in the original equation. This without resorting arcane methodologies that I really do not understand. Or need to for that matter.
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. ;)
Thank you for your channel. I always loved math and now use your channel for brain exercises (I am 74). Shoutout to all my math teachers. I went as far as calculus -
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.
So, I found the LCD for both x/4 and 8/x, which produced x/16 and 8/4x. Since the denominators much match, 16 and 4x must be equal, thus x=4. Does this work consistently?
I solved it in one sec, just looking at the thumbnail as the notification of the video appeared.
Second degree quadratic equation : ax^2 +bx +c = 0
Delta = b^2 - 4ac
X1 = (-b + √(Delta) ) / 2a
X2 = (-b - √(Delta) ) / 2a
So with x^2 - 12x +32 = 0
a=1, b=-12, c = 32
Delta = 16
X1 = 8
X2 = 4
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
Step 1 - multiply everything by 4x (the common denominator)
which leads to x^2 + 32 = 12x
Step 2 - put into standard quadratic form
which leads to x^2 - 12x +32 = 0
Step 3 - factor that into (x-8)(x-4) = 0
The two roots are x=8 and x=4.
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
X=4 ie 4x( x/4 +8/x) = 4x × 3
ie x.x + 32 = 12x
x.x + 32 - 12x = 0
X.X - 8X - 4X + 32= 0
X(X-8) - 4(X-8)=0
(X-8)(X-4=)=0
Hence X = 8 or 4
Loved this. I also use the box and X method
Mental math 101. Look, think, solved it! Something + something = 3. Answer: 4 😊
Or 8, it's a quadratic
Correct. Because 4/4 + 8/4 = 1 + 2 = 3
(x/4)+(8/x)=3
=>(x^2+32)/4x=3
=>x^2+32=12x
=>x^2-12x+32=0
=>x^2-4x-8x+32=0
=>x(x-4)-8(x-4)=0
=>(x-4)(x-8)=0
=> x=4,8
I didn't bother to watch more than the first few minutes. It takes about a minute to solve it
As an aspiring woodworker, I deal with fractions everyday. As soon as I looked at this, I knew that x=4 because 4/4 (four quarters)=1. 8/4 (eight quarters)=2. Wood that is four quarters thick is 1 inch. Wood that is 8 quarters thick is 2 inches. 1"+2"=3" or, 1+2=3. Then you said there could be multiple answers and I instantly realized that 8/4 (eight quarters)=2". 8/8 (eight eighths - silly way of writing it, for woodworking, that is - was the same as four quarters)=1". 2"+1"=3" or 2+1=3. So, x=4 OR x=8. I did all the first calculation in my head in about 10 seconds. The second one, once I knew to look for it, was nearly instantaneous. There absolutely was no need for tangent after tangent and tangents running off OTHER tangents, talking about polynomials and FOIL and other things that I don't think I ever learned in my Introductory to Algebra class in 9th grade (which was 44 years, ago) and Algebra 1 class in 10th grade (43 years, ago), both taught by two different teachers. Both of them had one other thing in common (besides being bald-headed math teachers) - they each had the personality of a piece of wood and would drone on and on putting most students to sleep (think Ben Stein from Ferris Bueller's Day Off). Your video, sans the lack of personality, brought me right back to those two classes and I immediately started to doze off. At the very least, my mind started wondering. I'm actually amazed I made it through to the end as I was about to skip on to the next video which is going to be FAR more interesting - it's about space and stuff. I realize you need to make these at least 8 minutes, in length, to get paid by YT but, even 8 minutes is really stretching this lesson beyond most people's ability to stay focused. Just try it with less tangents...you might get better feedback. Just saying...
Isn’t it 4 without even any solving? 😂😂
Yeah 4 is my def answer
After 50+ years of taking algebra, I got this with no problem. Easy peasy.
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.
He is teaching. For those of you who know it already, please be patient with him and those who are trying to learn it. Thank you.
Great content. How about just 1 tangent per video?
Why does it take you 20 minutes to solve a problem that I can do in my head in 30 seconds???
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.
I figured it would have to be 4, but never thought of 8 also being a correct answer.
I didn't have to do any of the factoial stuff, but i can understan that more complex larger rational expressions would require the technique, so thank you for explaining the process.
x1 = 4, x2= 8. Quadratic formula, sum and product of roots, or factorization.
For some reason I arrived at 4 and 8 but differently. I moved the 8/x to the other side resulting in x/4 = 3 - 8/x, which becomes x/4 = (3x - 8)/x. Cross-multiplying gets me x^2 = 12x - 32, which becomes the same quadratic equation you have of x^2 - 12x + 32 = 0. Factored out that leads to 4 and 8.
why applying a non-mathematical 'trick' such as cross multiplying .
you could just bring all terms to the left and over a common denominator being 4x,
and ask : when is a fraction equal to zero ?
answer: if the numerator =0 And the denominator ≠0 .
so 4x ≠ 0 => x ≠ 0 ,
And at the same time
x² - 12x + 32 = 0 .
completing the square :
x² + 2·(-6x) + (-6)² + 32 = 0 + (-6)²
(x - 6)² = 36 - 32
x - 6 = ± √4
x = 6 ± 2 .
both possibilities correspond to x ≠ 0 .
so x = 8 or x = 4 .
What you are seeing is the schtick. He is an elite mathematician and wants to give you a real math experience. The "prollem" looks simple and turns out it is rife with pitfalls. You, therefore, can't solve it and he steps in step by step and solves it. This is how you learn math. If you want to play childhood T-ball, this is not the guy. He's gonna test you and, frankly, I like that. He also designs tests for a living, so he's going to know how folks get sidetracked. That is valuable information! Love, love, love this dude!
The general case is:
(x / a) + (na / x) = (n + 1)
That can be solved by applying this formula:
x = (a / 2) [(n + 1) ± (n - 1)]
Which is:
S: { a ; na }
Does it require so much time to solve/explain this?
I do agree with those who are saying that there is too much unrelated side talking which makes me to give up on the video before it ends....... the content is great but it's annoying to listen!!!! Sorry but that's my honest opinion!!! I gave it a like though to support this kind of videos that are actually helpful!!!
x/4 + 8/x = 3
Let t = x/4
t + 2/t = 3
t^2 -3 t + 2 = 0
(t -1)(t-2) = 0
x = 4 t
x = 4
Or
x = 8
This is the way I did it and it seems to simplify things. Noticing the 2nd term is 2/t if the first is t is a nice simplification.
formula for roots of a quadratic equation X**2 -12X + 32 = 0 (which we get when multiplying each element with 4X) gives us 8 and 4 as 2 roots of X
So what happens if you multiply both sides by 4x?
You get
x^2 + 32 = 12x
Take note that 0 is not a valid solution to our original equation.
We can rearrange to form:
x^2 - 12x + 32 = 0
Now we can factor or use the quadratic formula
Factoring easily yields:
(x-8)(x-4) = 0 giving us the solutions 4 and 8
(Using +- as the symbol for "plus or minus")
The quadratic formula yields
(12+-sqrt(144-4*32))/2 =
6+-sqrt(144-128)/2 =
6+-sqrt(16)/2 =
6+-4/2 =
6+-2
Which again gives us 4 and 8 as solutions.
Unlike many of the other comments, I didn't watch this video but like everyone else, I am curious how you managed to drag out a 30 second to 2 minute problem into 20 minutes.
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.
Excellent video!
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!
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.
Thanks for the video
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.
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).
I am not sure, what is this about? it is a 10 seconds problem. x^2 + 32 = 12x (note x/= 0). x^2 -12x +32 = 0. (x-4)*(x-8) = 0. {x=4;x=8} It is longer to write then to solve it in ind, w/o paper!
Thank you John, I appreciate your content.
The easiest and BEST way with answer is @18:00 mark. Simple and takes 10 seconds to get the answer instead of 20 minutes.
4x(x/4 + 8/x) = 3
X^2 + 32 = 12x
X^2 - 12x + 32 = 0
X^2 -8x - 4x + 32 = 0
X(x -8) - 4( x- 8) = 0
(X -4)(x - 8) = 0
X = 8 or x = 4 ->
I multipled both sides of the equation by x to get the variable out of the denominator. Then you get a quadratic equation. I moved everything to one side of the equation so it’s equal to 0. Then I factored to get (x-8)(x-4)=0. So the solutions are 8 and 4.
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.
So I switched denominators did the fractions and well x over x = 1 : 8 over 4 = 2 so 1+2= 3 : x/4 or x/8 is always = to 1 because both answers will be 4/4 or 8/8
I came up with 4 in a matter of seconds, when you said two solutions I came up with 8 quickly
I'm going with x=4 or 8, both work. ???
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.
what took him 20 minutes to solve took me 2 seconds.
Took Algebra 1 in 1969. I got "C"s because I didn't show my work. I never learned how to do the work. I got the answers correct. Teacher accused me of cheating of my neighbor. He showed his work but got the wrong answers.
What a wicked world!
Because of algebra I know that x will be the same number and that there might be more than one choice. Basic knowledge of fractions told me that if I could get fractions equal to 1 and 2 it would work because 1+2=3 and 2+1=3. 4/4=1 and 8/4=2 so one solution is 4. 4/8=2 and 8/8=1 Therefore the second solution is 8.
Shalom to everyone
Does the teacher John have work sheets. About 100 drrills per sheets ?
Thanks.
I was always bad in maths but I could solve it in less than 10 seconds just using my head
辺々4xを掛けx^2+32=12x
12xを移項し(x-4)(x-8)=0
∴x=4,8
Thsnk you! Thank you! Thank you! ❤