For equations of the form √(a-x) = a-x² and assuming |x| ≤ min(a, √a) we can square both sides: (a-x) = (a-x²)² = a²-2x²+x⁴. Now look at this as an *equation in a* , not in x: a²-(2x²+1)a+(x⁴+x) = 0. And this miraculously factors because D = (2x²+1)²-4(x⁴+x) = 4x²-4x+1 = (2x-1)², so we get a = ((2x²+1)±(2x-1))/2 from where we get just two quadratics in x. The equation in the video corresponds to a = 7 case.
@leif1075
2 жыл бұрын
Where did you get that derivation from..where did the a go?
@MyOneFiftiethOfADollar
2 жыл бұрын
Very nice substitution with a. Also the “miraculous factorization” to be expected due to two intersection points of hand sketched graphing transformations of the parabola and square root function. Despite the deceptive title “simple trick solves in seconds”, very enjoyable video that also illustrates the value of restricting the solution domain to find the positive solution
@andrewphilip3308
2 жыл бұрын
Thank you, and the two quadratics give us the four points of intersection of two parabolas y^2 = a -x and x^2= a - y which yield a nice symmetrical pattern when graphed.
@willbishop1355
2 жыл бұрын
Neat trick, although this problem is not hard to solve directly. The rational root theorem quickly shows that -2 and 3 are roots of the quartic, and from there you have the quadratic x^2 + x -7 = 0 and check each solution as shown in the video.
@bach556
2 жыл бұрын
Video Title : a simple trick that solves this equation in SECONDS. Video Length : 10 MINUTES
@tomansager1
2 жыл бұрын
Me: 3
@taharanjbar3679
2 жыл бұрын
😀 exactly.
@wernergamper6200
2 жыл бұрын
Solves in seconds, yet the video is >10 min long?
@wjx8439
2 жыл бұрын
what i noticed is that you can subtract the two equations x = 7 - y^2 and y = 7 - x^2. => y - x = y^2 - x^2 => y - x = (y - x)(y + x) => (y - x)(y + x - 1) = 0 => (7 - x^2 - x)(7 - x^2 + x - 1) = 0 => (x^2 + x - 7)(x^2 - x - 6) = 0 and you can check the condition that whether x
@createyourownfuture5410
2 жыл бұрын
How do you go from step 2 to 3?
@wjx8439
2 жыл бұрын
move the y - x to the RHS and factor it out
@a.f.5844
2 жыл бұрын
Starting from y-x=(y-x)(y+x), either y-x=0, I.e. y=x or y+x=1, I.e. y=1-x. If y=x then from x=7-y^2 -> x^2+x-7=0. If y=1-x -> x^2-x-6=0. From each of these square equations only one solution is valid.
@createyourownfuture5410
2 жыл бұрын
@@wjx8439 oh OK, thanks
@traianosyioultsis6721
2 жыл бұрын
Another straightforward trick to solve the equation sqrt(a - x) = a - x^2 is to substitute a - x^2 = x + u. The reason to do this is that a - x = x^2 + u and the original equation becomes sqrt(x^2 + u) = x + u, in which the x^2 terms cancel each other if it is squared: x^2 + u = x^2 + 2xu + u^2, hence u^2 + 2xu - u = 0. This gives two quadratic equations, u = 0 or x^2 + x - a = 0 and u + 2x -1 = 0 or x^2 - x + 1 - a = 0. You get all four solutions to the underlying quartic equation, even if the rational root theorem does not provide any easy root (check, for instance, the case a = 5). Of course, you have to check if abs(x)
@danilonascimentorj
2 жыл бұрын
say sqrt(7-x)=y and then we have the system of equations x=7-y^2 and y=7-x^2; subtracting both equations we have either x=y or x+y=1; First gives (-1+sqrt(29))/2 and second x=-2.
@pageegap
2 жыл бұрын
Very nice and quick way to solve !
@thefreeze6023
9 ай бұрын
√(7 - x) = 7 - x^2 x^2 = 7 - √(7 - x) Take the square root of both sides. x = √(7 - √(7 - x)) Replace the 'x' in the RHS with the entire RHS. x = √(7 - √(7 - √(7 - √(7 - x)))) This step can be repeated infinitely. Thus: x = √(7 - √(7 - √(7 - √(7 - ...)))) Therefore: x = √(7 - x) Square both sides x^2 = 7 - x Rearrange x^2 + x - 7 = 0 Solve the quadratic: x = (-1 + sqrt(29)) / 2 = 2.193, x = (-1 - sqrt(29)) / 2 = -3.193 Check these two solutions. Notice that x = -3.193 is a false root, since it leads to a negative on the RHS of the original equation, and the LHS is a square root which cannot be negative. Thus x = (-1 + sqrt(29)) / 2 = 2.193
@e.9443
5 ай бұрын
I love this one
@tianqilong8366
2 жыл бұрын
yea, directly solve the equation is too boring, nice method!
@changjeffreysinto3872
2 жыл бұрын
Lmao IMO Prelim x=sqrt(7-sqrt(7-x)) x=f(f(x)) where f(x)=sqrt(7-x) Since it is clearly invertible, and the intersection will be where x=sqrt(7-x)
@petersievert6830
2 жыл бұрын
But -2 cannot be a solution, as x>0 was needed or did I miss something?
@taharanjbar3679
2 жыл бұрын
Hi. see again clip from 6:40.he said in general case not x>0.
@petersievert6830
2 жыл бұрын
@@taharanjbar3679 Ah okay, thank you. I fast forwarded to the end to simply check my solution.
@fix5072
2 жыл бұрын
You could also write x as the limit of the series x_(n+1)=sqrt(7-x_n). We know the limit exists so we just have to solve x^2+x-7=0
@muhendisgenc8216
2 жыл бұрын
yep
@khiemngo1098
Жыл бұрын
Hello, I greatly enjoyed your video. Many thanks! By the way, I still don't understand why we have to reject x = 3 for the general case when there is no restriction on x. For the general case, when x = 3, RHS = -2 while LHS = sqrt(7 - 3) = +2 OR -2. So for x = 3, we do have the case when both sides = -2. I'll appreciate your clarification! Thanks.
@wasimvillidad3000
2 жыл бұрын
Not wishing to burst your bubble but your trick misses two positive solutions with this example: y = √(7-3x) and y = (7 - x²)/3.
@fdr2275
2 жыл бұрын
The simple trick is to use an electronic calculator and the solution will come out in a matter of a second. You can spend a lot of time to play with this kind of math manipulation but in reality and for all practical purposes, people would just plug it into a cheap electronic calculator to get the answer. 60 years ago, you might see a lot of this kind of problem on the university entrance exams. Today, you are not going to see it on the AP exams.
@panos1435
2 жыл бұрын
Awesome solution! Keep up with this!
@iainfulton3781
2 жыл бұрын
All results are valid cause square roots can be negative. Turn off postifications
@babaji1947
2 жыл бұрын
so it takes 10 minutes to solve an equation in seconds???
@danilonascimentorj
2 жыл бұрын
you said in the beginning that x>0, so why you found x=-2 as a solution?
@WerewolfLord
2 жыл бұрын
At 6:46, he removed the restriction of x>0.
@leif1075
2 жыл бұрын
@@WerewolfLord why if he started with it..he has to finish with it.
@Trixex
2 жыл бұрын
@@leif1075 he showed the trick you can use with the restriction and then proceded to show what to do in case there wasn't. It's like a bonus.
@Trizzer89
2 жыл бұрын
This took a bit longer than seconds
@ananyagupta1409
2 жыл бұрын
Nice!
@Ayush-gv5ow
2 жыл бұрын
jee student ho ka
@alanjones4358
2 жыл бұрын
Can you explain why you rejected x = 3? Why can't 7 - x^2 be negative?
@reyanshsharma1312
2 жыл бұрын
because square roots cannot be negative, and so -2 cannot be a solution to a square root
@alanjones4358
2 жыл бұрын
@@reyanshsharma1312 Since when? The last I checked, 4 has two square roots: 2 and -2. And it's considered a mistake to forget to include both when solving a problem.
@renka-chan9213
Жыл бұрын
@@alanjones4358 They were reffering to the principal square root, which Is, according to its range, is always positive.
@alanjones4358
Жыл бұрын
@@renka-chan9213 I don't think they specified that in the problem. The equation as presented has two real roots, why reject one?
@renka-chan9213
Жыл бұрын
@@alanjones4358 they didn't. That's because it was already taken as "given information". The "always positive" rule is enforced in, like, every problem involving square roots.... For instance, let's say that I want to compute sqrt(576). What do you think the answer would be?
@LivYi
2 жыл бұрын
Nice! Any real world example that brings about such an equation?
@andrewphilip3308
2 жыл бұрын
So you think this is not the real world|? Perhaps it's the other way round. At all events this world makes me feel happy and the othe one usually makes me feel miserable.
@atpugnes
2 жыл бұрын
Since 7-x^2 is an even function shouldn't there be another solution for y=-x, which is (1-sqrt(29))/2?
@sebastianw.1217
2 жыл бұрын
As per the beginning we are only considering positive x, so while that is an intersection we aren't interested in it.
@leif1075
2 жыл бұрын
Isn't the only solution x equals 3?
@jkid1134
2 жыл бұрын
I don't like this title. 10 minute video promises seconds-long trick, no matter how you slice it something doesn't add up
@andrewphilip3308
2 жыл бұрын
600 seconds long!
@jkid1134
2 жыл бұрын
@@andrewphilip3308 Eh, alright. It's just, the way I think it poises me to be impatient is maybe counterproductive to retention.
@andrewphilip3308
2 жыл бұрын
@@jkid1134 Only joking really. One seldom sees a youtube video title which has a close correspondence to the contents. :)
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