This method is far too complicated. Express y=48/x, substitute in x²-y²=28. Let t=x² and solve. t²-28t-48²=0

@mas_logic

6 ай бұрын

David! Thank you for your comment and suggestion! Your approach is indeed a valid alternative for solving the problem, and different methods can provide varying perspectives. The method I chose, using the difference of squares formula, is just one way to approach the problem. It aims to showcase a particular algebraic technique that can be useful in other scenarios as well. Your suggestion of expressing \(y\) in terms of \(x\) and substituting it into the equation is another valid and efficient approach, especially in this specific case. It's great to have different methods in our toolkit for problem-solving. I appreciate your input, and I encourage viewers to explore various approaches to find the one that resonates best with them. If you have any more questions or if there's another problem you'd like me to tackle in a different way, feel free to let me know!

@God-gi9iu

6 ай бұрын

@@mas_logic😮

@GaryBricaultLive

4 ай бұрын

I totally agree with you. Why turn a 2 minute solution into a 13 minute oration.

@mattslupek7988

4 ай бұрын

(x + y)(x - y) = 28 Find factors for 28 Use 14 x 2 as factors x + y = 14; x - y = 2 Cancel out the y’s Divide (14 + 2) by 2 2x = 16 x = 8 8 + ? = 14; 8 - ? = 2 y = 6 8 x 6 = 48 8^2 - 6^2 = ? 64 - 36 = 28 Voilà!

@m2indurkar

3 ай бұрын

(x+y)(x-y) =2*14, Then x+y =14 ...(1)and x-y =2..(2) Adding both eqn 2x =16 ,x=8 and y=6 .Let verify this value with eqn x.y=48, 8.6=48. Hence assumed 2factor , x+y =14 and x-y =2 is valid.

(x-y) x (x+y) = 28 (x-y) x (x+y)= 14 X 2 x-y=2 x+y=14 2x=16 ============> x=8 ===========> y=6 x.y=48 ============> 8 x 6= 48 x+y = 8+6= 14. Mauro Moreira (Brasil)

@JangirBK

Ай бұрын

Why 14×2 only..? Why 28×1 and 7×4 left ??

x²-y²=28 xy=48, y=48/x Substitute in x²-y²=28 x²-(48/x)²=28 x²-48²/x²=28 Clear fraction (x²)²-48=28x² Let x²=t t²-28t-48²=0 Complete the square t²-28t+(28/2)²=48²+(28/2)² t²-28t+14²=48²+14² (t-14)²=(50-2)²+(10+4)² (t-14)²=2500 t-14=√2500=±50 t=14±50 t=64, -36 Back substitute x²=t=64 x=±8 x²=t=-36 x=±6i we can reject this complex solution When x=±8, y=±6 Checking (±8)²-(±6)²=28, (±8)*(±6)=48 Finally x+y=+14, -14

@mas_logic

5 ай бұрын

David, it is great! I love your method.

Hints: 64 - 36 = 28 and 8 x 6 = 48. Negatives work also since we're squaring and multiplying.

@mas_logic

6 ай бұрын

Fantastic! It's wonderful to see collaboration and different problem-solving approaches in our community.

@l.w.paradis2108

6 ай бұрын

What I did, too. Then the video is interesting for demonstrating other methods.

@tedn6855

5 ай бұрын

Negatives and imaginary also work y =8j and x=-6j

Thre's no way I can come up with the algebra in this solution so I just use the most direct method. x^2-(48/x)^2=28 Let t=x^2 t^2-28t-2304=0 t=64, or -36 (rejected) x=8 or -8 x=8,y=6 or x=-8,y=-6 x+y=14 or -14

@mas_logic

5 ай бұрын

Stephen! I love your methodology.

x^2-y^2=28 --> (x-y)(x+y)=28 (spoiler won't be much use) xy=48 --> y=48/x So x^2-(48/x)^2=28 --> x^4-48^2=28x^2 --z=x^2--> z^2-28z-48^2=0 --> (z-14)^2-2500=0 --> z=14±50 --> x^2=64 or -36 --> x=±8 or ±6i, --> y=±6 or ∓8i --> x+y=±14 or ∓2i.

Not sure why the imaginary roots were discarded! Nowhere in the problem description does it say that x, y, etc., are real. This problem has 4 solutions: x=8, y=6; x=-8, =-6; x=0+6i, y=0-8i; x=0-6i, y=0+8i. accordingly x+y is +14, -14, 0+2i, or 0-2i.

@mas_logic

5 ай бұрын

Dear, you are correct! To make it short, I have solved the problem only for real values of x.

Why such a simple question needs 13 mins to solve????????

Should also verify the solutions (in case one would be rejected) If x + y = 14 Then x - y = (x2 - y2)/(x+y) = 28/14 = 2 Solving for x & y, we get x = 8, y = 6 So, xy = 48 (Holds true) If x + y = -14 Then, x - y = -2 Solving for x & y, x = -8, y = -6 So, xy = 48 (Also holds true) Hence both 14 and -14 are accepted as solutions. Interestijg that we have two different (x, y) pairs satisfying the equations.

@mas_logic

6 ай бұрын

Very Nice! In next videos, I will also verify the solutions. Thanks for your suggestion.

@l.w.paradis2108

6 ай бұрын

No surprise: negative times negative equals positive

Nice problem, but you made it too complicated 😢

Xis 8 and Y is 6 .From very sight of the problem itself I can say. This method is known as Induction method.

@mas_logic

4 ай бұрын

Great 👍

On a timed test a straight line approach to a solution is preferred.

@vasilispoulikakos7512

3 ай бұрын

x+y=14

It is just a difference of two squares... X^2 - y^2 = 28 (X+y) (x-y) = 28 Factor the 28, so (X+y)(x-y)= 14(2) then (X+y) = 14 and (x-y)=2 to check if you have chosen the correct factors 14 & 2, (X+y)(x-y) = (8+6)(8-6) Then (X+y)= (8+6)= 14 true (X-y)= (8-6)= 2 true

@mas_logic

5 ай бұрын

great

Easy method Let x^2 =a and y^2= b So a - b = 28 ab = 48^2 so find (a+b) using identity thus x^+y^2 and add 2xy to get (x+y)"2 and hence x+y

I normally solve the problems using the extensive long method like you do, and l still do anyhow, but l try to eliminate it as much as possible... Heed, when you reach: (x+y)^2 - 784/(x+y)^2 =192, to simplify things, you should have let either x+y = a or (x+y)^2 = a... pick your choice, the equation would have more likely become much simpler.... the use of substitution, l tend to make the same mistake also at times...

@mas_logic

5 ай бұрын

Great Suggestion

@ilhamibingol3673

26 күн бұрын

14

X=8 Y=6 X*Y=48 X^2 - Y^2= 8^2 - 6^2= 64- 36=28

This is very simple just guess work using factors of 48

I just used prime factorization and found x + y = 14 immediately, but did not know whether the solution was unique. The initial equation, the difference of squares, was a hint that it was not.

@mas_logic

6 ай бұрын

Dear! The solution is not unique for you, because you are unique 💕. Thanks for the feedback.

@dubinkaperelmana

5 ай бұрын

You can't use prime factorization in general, because the system of equations was given not exactly in integers

thank you for a wonderful video, suggestion: why not substitute x plus y all squared by z?

@mas_logic

5 ай бұрын

Great suggestion! Thanks for the feedback.

14

Hi, I factored (x^2-y^2) with (x+y)(x-y)=7*4 or 14*2. 7*4 didn't rule but 14*2 is fine. x+y=14 and x-y=2 therefore x=8 and y=6. 8 and 6 agree with the second condition that xy=48…. fast, smart but not as elegant as yours

@mas_logic

4 ай бұрын

Thanks Dear! Your words encourage me to continue the work.

This one was solved quickly... I just said to myself: 48..could the solution be 6 and 8. Oh, it is! :)

@mas_logic

6 ай бұрын

It's Great!

If this an Olympiad question surely myself and many would qualify for that....but it never happened.

Of 48

Since you started with a difference of two squares and you were looking for the value of 'x+y', you should start with the statement that your solution MUST be a factor of 28. Also, your answer 'x + y = +/-14' is INCOMPLETE since you NEVER addressed the condition that xy=48!

@user-vx4lb8rj8r

2 ай бұрын

but he did address the condition that XY equals 48 in the very beginning when he substituted 4ab in for 4xy and put 48 in for XY..., thus making the equation=196 2:09

Very nice

@mas_logic

4 ай бұрын

Thanks

Very good analysis of the problem

@mas_logic

5 ай бұрын

Thanks

Actually 8j and -6j also works which means 2j is also an answer

@mas_logic

5 ай бұрын

What is "j"?

@tedn6855

5 ай бұрын

@@mas_logic others call it i

@tedn6855

5 ай бұрын

Engineers use j instead of I for square root negative one

X is 8 y is 6 . Elementary question . Can be solved orally

Using complex analysis method guys

x=8 e y=6, portanto x2=64 e y2=36. Temos x2-y2=28 e xy=48. 64-36=28, portanto 8+6=14. É possível fazer esse cálculo de cabeça (raciocínio lógico)!

(8; 6); (-8;-6) on peut dire x+y=14 ou x+y=-14

❌ = 8 , 🌱 = 6

Davidtallent предложил оптимальное решение. Зачем ломать дверь,когда в кармане есть ключ?

@mas_logic

5 ай бұрын

Что делать, если к одному пункту назначения ведет множество дверей? У меня был ключ от другой двери. Спасибо за ваш комментарий.

Why reject the complex roots?

@mas_logic

4 ай бұрын

just to make video shorter

Hi. Even if there are other comments describing how the solution for this olympiad was found, I will leave my own comment anyway. X=8 & Y=6. 8 to the 2nd power is 64 and 6 to the 2nd power is 36. Therefore, 64-36=28. 8*6=48 and 8+6=14. It's all as simple as that. As mentioned before, if anyone else has already come up with this same solution, I will just leave these comments anyway. Enjoy using the values I was able to find for both X & Y to solve all these equations.

-14,-2,2,14

6, 8. 14

x+y=14

Why so much calculation?just put y=mx,divide both of them,solve 12m^2+7m-12=0 bit more and we get answer.

@mas_logic

4 ай бұрын

I love your intelligence 💕

x+iy

X-Y multiple X+Y

What is this method? Simplest way is there.

Its too. Long answer i

14.

x² = y² + 28; (xy)² = x²y² = 48² ; ---> (y² + 28)y² = 48² ; （y²）² + 28 y² - 64* 36 =0 ; y² = 6 .....

@mas_logic

5 ай бұрын

Looks Good!

Blud fr spawned in 784 to 24💀💀 Even said that 192 = 196 + 4💀💀💀💀💀

@mas_logic

6 ай бұрын

Hi there! I'm curious about your comment. Could you please provide more context.

@ratty_robloxian

6 ай бұрын

@@mas_logicwhy did you change from 24 to 784 for no reason, and said that 192= 196+4

@mas_logic

6 ай бұрын

@@ratty_robloxian -192 = -196+4

@ratty_robloxian

6 ай бұрын

@@mas_logic thanks but what about the 24 to 784

@mas_logic

6 ай бұрын

@@ratty_robloxian Dear, Please tell me the timestamp where I did that so that I can tell you about that.

There is much easier way to resolve the problem

x^2-y^2=28 xy=48 x+y? (x^2-y^2)^2=28^2 x^4+y^4-2(xy)^2=784 x^4+y^4-4608=784 x^4+y^4=5392 (x^2+y^2)^2-2(xy)^2=5392 (x^2+y^2)^2-4608=5392 (x^2+y^2)^2=10000 |x^2+y^2|=100 x^2+y^2=-100 (x+y)^2-2xy=-100 (x+y)^2-96=-100 (x+y)^2=-4 |x+y|=2i x+y=[+-]2i ❤❤ x^2+y^2=100 (x+y)^2-2xy=100 (x+y)^2-96=100 (x+y)^2=196 |x+y|=14 x+y=[+-]14 ❤❤

@mas_logic

6 ай бұрын

Kudos to ChavoMysterio for bringing a fresh perspective to the problem! Your thorough explanation and the clarity in your algebraic steps really stood out. It's fantastic to witness the variety of approaches to the same mathematical challenge. Well done!

Corretion. X=+-8 Y=+-6

@rabotaakk-nw9nm

4 ай бұрын

The task question was "x+y=?". The answer is x+y=±14

@ceciliafreitas7856

3 ай бұрын

@@rabotaakk-nw9nm 8+6=14

@ceciliafreitas7856

3 ай бұрын

@@rabotaakk-nw9nm Read my corretion.

We have the following two equations: 1) x^2 - y^2 = 28 2) xy = 48 We can solve this system of equations by first solving the first equation for x^2: x^2 = y^2 + 28 Now, we can substitute this expression for x^2 into the second equation: (y^2 + 28) * y = 48 Distribute the y^2 term: y^3 + 28y = 48 Now, we can factor out y from the equation: y(y^2 + 28) = 48 Next, we can divide both sides by (y^2 + 28) to isolate y: y = 48 / (y^2 + 28) Since we have y in the denominator, we need to find the value of y that makes the expression in the denominator zero. This will give us a quadratic equation: y^2 + 28 = 48 Subtract 28 from both sides: y^2 = 20 Now, find the square root of both sides: y = sqrt(20) y = 2 * sqrt(5) Now that we have the value of y, we can substitute it back into the expression for x^2: x^2 = (2 * sqrt(5))^2 + 28 x^2 = 8 * 5 + 28 x^2 = 40 + 28 x^2 = 68 Finally, find the square root of both sides to get the value of x: x = sqrt(68) Now that we have both x and y, we can find the sum x + y: x + y = sqrt(68) + (2 * sqrt(5)) Using a calculator, we find that: x + y ≈ 11.29

@user-vx4lb8rj8r

2 ай бұрын

in the e like second step where you said substitute this expression for x² into the second equation... the second equation is just XY=48. not x²y=48 you would need to take the square root of both sides and then put that value in for x. pass that your entire thing is wrong. try again though.

@karimlachboura7276

2 ай бұрын

@@user-vx4lb8rj8r yes you ´re right The solution goes like We can solve this system of equations for x and y by recognizing it involves the difference of squares factorization. Here's how we proceed: Difference of Squares Factorization: The equation x^2 - y^2 = 28 can be rewritten using the difference of squares pattern a^2 - b^2 = (a + b)(a - b) where a = x and b = y. In this case, we have: (x + y)(x - y) = 28 Incorporate the Second Equation: We are also given xy = 48. We can use this information to express one variable in terms of the other. Since we're aiming to eliminate y, let's solve the second equation for y: y = 48 / x Substitute y in the First Equation: Now, substitute this expression for y in the first equation obtained from the difference of squares factorization: (x + (48 / x))(x - (48 / x)) = 28 Simplify and Solve: Expand the brackets in the resulting equation: x^2 + 48 - (48 / x) - x^2 = 28 48 - (48 / x) = 28 Subtract 28 from both sides: (48 / x) = 0 This equation leads to a contradiction because dividing by x cannot result in zero. Therefore, there are no real number solutions for x and y that satisfy both equations simultaneously.

Xis 8 and Y is 6

Too long method ! Better proceeding like this : Let X^2 =a and -X^2 =b and s=a+b .then ab=-(XY)^2 and using the equation t^2-st+p=0

@mas_logic

6 ай бұрын

Nice approach dear! I will try in next videos. Thanks for the feedback.

Couldn't you just check the divisors of 28?

@mas_logic

3 ай бұрын

Heart for your suggestion ❤️

X is8and Y is 6

X=7 y=4 49-16=28 7x4=28

@ThorsHammer1

4 ай бұрын

how is 49-16=28 EVER TRUE?!

@rabotaakk-nw9nm

4 ай бұрын

And 7×4=48 ???

@ceciliafreitas7856

4 ай бұрын

@@rabotaakk-nw9nm You didn't read well. I wrote 4 x7=28 not 48. Ok. Read write. 🇵🇹

@ceciliafreitas7856

4 ай бұрын

@@ThorsHammer1 It was a distraction. Sorry. 🇵🇹

@rabotaakk-nw9nm

4 ай бұрын

@@ceciliafreitas7856 xy=48 !!!

you steal from pre-math

@mas_logic

5 ай бұрын

You're right, math problems are universal. I take pride in creating original content, and any similarities are purely coincidental. If you have specific concerns or suggestions, feel free to let me know, and I'll address them.

X+Y =14

(X + Y) * (X - Y) = 28 We have : 28 * 1 ; 14 * 2 ; 7 * 4 . Only 14 * 2 works X + Y = 14 X - Y = 2 , them 2X = 16 ; X = 8 and Y = 6 . Bingo from Brazil !!!!

Done in some 30 sec. The authors of this kind of puzzle profit from the conjuncture that they know the algorithm to create pythagorean triples which ordenary people don't. Given n^2 - m^2 = a and 2nm = b, n^2 + m^2 = c = sqrt(a^2 + b^2). In this case we can cancel 4 out of the first two numbers 28 and 96 = 2xy, so we get 7 and 24 as the legs of the right triangle. As 7 is small in comparison to 24, we guess that 25 is the hypothenuse. The difference between 24^2 and 25^2 is of course 49 (24 + 25), so 7 is the 2nd leg. Multiplied with the cancelled 4 we geht x = 96 and y = 28. So x + y = 124. Shoddy trick, poor juggler.

@mas_logic

6 ай бұрын

My aim is not to trick or fool anyone. What do you want to suggest for the next videos?

I don't know why he opted for the longest possible method 😅😅 I think dude wants as much watch time to his credits as possible. Just use the substitution y=48x....

Sa. Jhaane tarkkaa pasa nd nahi aayaa hai mai n ise short main samjhaa doongaa

Easy x=8 y=6

@user-pd9le4bc7e

2 ай бұрын

Find the factors of 48 48 = 2 x 24 4 x 12 8 X 6 1 6 x 3 out of which difference of squares 28are 8 and 6 Sum = 8+6 = 14

Too long and complicated solve.Theoriclly correct.

1️⃣➗2️⃣：（X^2-Y^2）/（XY）=7/12，12X^2-7XY-12Y^2=0，（4X+3Y）（3X-4Y）=0， XY=12>0, 4X+3Y!=0, 3X-4Y=0, (X,Y)=(8,6),(-8,-6), X+Y=14,-14

@mas_logic

5 ай бұрын

Nice way! I like it.

İt's rather long.

Method too long

@mas_logic

5 ай бұрын

Do you have the shorter one?

Х+у равно 14

Ужасное дерганье рукой, из-за этого трудно смотреть. Постоянно выделываются какие-то пассы рукой без малейшей необходимости. Будьте спокойней, когда записываете ролик, и качество материала существенно улучшится.

@mas_logic

3 ай бұрын

Мой дорогой друг, мне нравится твое предложение. Благодаря вашей конструктивной критике я совершенствуюсь с каждым днем. Мне нравится ваш комментарий.

What are you doing

@mas_logic

6 ай бұрын

Solving the problem, Sir!

@user-cg5xv4zz2b

6 ай бұрын

the problem is that I have is that I do not know why you put down the 2 equations - (a=b)^2 = ... and (a-b)^2 = ... I have no clue - tried to watch - do not seem to be able to see why and so I see no logic in solution. I assume that other person is in same boat. @@mas_logic

@mas_logic

6 ай бұрын

Your Question: "Why did you include the expressions (a+b)^2 and (a-b)^2 in the solution? What's the logic behind this?" Dear, The inclusion of (a+b)^2 and (a-b)^2 is part of a technique called the difference of squares formula. This formula is crucial in simplifying expressions involving squares. The difference of squares formula states that for any values of a and b: (a+b)^2 - (a-b)^2 = 4ab In the context of this specific problem, we utilized this formula to simplify the expression (x+y)^2 - (x-y)^2. By doing so, we transformed the expression into 4xy, which allowed us to work with the given information more efficiently. It's a common algebraic technique used to simplify expressions and solve equations involving squares. If you have further questions or if there's a specific part you'd like more clarification on, feel free to ask!

@user-cg5xv4zz2b

6 ай бұрын

this was problem - x^2 - y^2 = 28 xy = 48 find x + y so how do you go from x^2 - Y^2 = 28 to the (x+y)^2 - (x-y)^2 - there is a logic jump. I saw the math to get 4AB ( or 4XY ) - but the jump from problem stated to putting in 2 equations that seem to be unrelated is the jump I do not understand. I @@mas_logic

@mas_logic

5 ай бұрын

As it is said, everything is fair in Love and War. I say everything is fair in Mathematics until the new step taken can be reversed back to the original. ~Mas

This method is too complicated

Very inefficient.

Too long

YouRe unnessary lingerin

8 и 6. Моментально ❗

@rabotaakk-nw9nm

4 ай бұрын

🤣 No! x+y=14 & -14

@user-el3yn5gh4p

4 ай бұрын

@@rabotaakk-nw9nm зачем тогда жить?? 🥴

@rabotaakk-nw9nm

4 ай бұрын

@@user-el3yn5gh4p Земеля! Живи счастливо, но правильно!

X=8, y=6

14

14

14

14

14

14

14

14

14

14

@mas_logic

6 ай бұрын

14 & -14

14

@mas_logic

6 ай бұрын

14 and -14