You Should Try This Amazing Algebra Question | 2 Different Methods to Solve

You Should Try This Amazing Algebra Question | 2 Different Methods to Solve
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[@StaR-uw3dc]

Third method: x^2-y^2=9 and xy=3 => x^2=y^2+9 and x^2y^2=9 after substitution 1st eq into 2nd we get quadratic in terms of y^2: (y^2)^2+9y^2-9=0 with one positive solution: y^2 = (-9+3sqrt(13))/2 so x^2=(9+3sqrt(13))/2. Further x^2+y^2=3sqrt(13) and xy=3 => (x+y)^2 = 3sqrt(13) +2(3) and x+y = +/- sqrt(6+3sqrt(13)). PS. sqrt(117) = 3sqrt(13)

[@اممدنحمظ]

تمرين جميل جيد . شرح واضح مرتب. شكرا جزيلا لكم والله يحفظكم ويرعاكم ويحميكم جميعا. تحياتنا لكم من غزة فلسطين

[@MathBooster]

Thank you 😊

[@jim2376]

(3/y)^2 - y^2 = 9 is one way to start.

[@danieldepaula6930]

Sure. Doing that, you will find y² = (sqrt(117) - 9)/2 . After that, using x² = 9/y² , you will find x² = (sqrt(117) + 9)/2 . Finally, using xy = 3, we have x² + 2xy + y² = sqrt(117) + 6. Then, x + y = +/-sqrt(sqrt(117) + 6).

[@jim2376]

@@danieldepaula6930 You da man, Daniel!

[@danieldepaula6930]

@@jim2376 thanks! Just trying to help :)

[@ИгорьКулик-л2р]

Очень долго. Надо разложить разность квадратов, разделить на x-y обе части. Возвести обе части в квадрат. В левой части будет квадрат искомой суммы, а в правой в знаменателе после возведения в квадрат добавить и вычесть 2xy. Получим в знаменателе квадрат искомой суммы( x+y)#2 и минус4xy,а это 12. Вот и все получилось биквадратное уровнение относительно (x+y) Решается весь это пример в уме ровно за 5 минут

[@zahrasadathashemi1125]

@@danieldepaula6930 I went the same way , but can you please explain that when the y²=(sqrt(117)-9)/2 , how the 9/y² is equal to (sqrt(117)+9)/2 ?

[@mohamadshareati7796]

سلام. خیلی عالی و شیوا و روان. افرین برشما . 👏👏👏👏👏👏👏

[@constantinfedorov2307]

Ну u = x² и v = -y² - два корня уравнения t²-9t-9=0; потому что u+v=9 и uv=-9; => x² = 9/2+3√13/2; y² = 3√13/2-9/2; если очень хочется, можно извлечь корни и сложить. Добавлю для ясности - конечно, произведение xy должно быть положительным при выборе знаков корней, откуда получается два решения (а не четыре).

[@koutenshi69gou]

Great! I also do this method!

[@bakoudujar]

@@koutenshi69gou me too...😊😊

[@admnner21111]

Would you tell me how 36 & 81 in the root is coming please???144+ 4•81 /2 in square root= 36+81 in square root??? Please write the detail for me🤔🤔🤔

[@William-Sunderland]

I separated x in both equations, got y^4+9y^2-9=0, found y which gives four solutions by the cuadratic equation, two discarted cause they are imaginary solutions sqrt of negative numbers, so two solutions positive and negative, then found x in terms of y, two solutions too, positive and negative, and then found x+y which gave a much larger equation as solution but comparing with calculator both answers were the same x+y=4.1008 (aprox) and x+y=-4.1008 (aprox) for both solutions.

[@schtorm2006]

X=3/y это для начала 9/y²-y²=9 y⁴+9y²-9=0 для начала

[@ИгорьКулик-л2р]

Разложим разность квадратов x+y=t, делим оье части на( x-y) и аозводим обе части в квадрат. В правой части раскрываем квадрат раности прибавляем и вычитаем 2xy.получаем в знаменателе t#2-12. Все получилось просейшее биквадратное уровнеие с некрасивыми корнями. Ровно 5 минут. Тем более, что коэффициент перед t#4 равен a=1, а b четное можно воспоьзоваться упрощенной формуой нахождения корней квадратного уравнения. 5 минут

[@raxz4008]

Another solution involving complex numbers is: Let i be the imaginary unit, so: (x + iy)^2 = x^2 - y^2 + 2ixy = 9 + 6i (x - iy)^2 = 9 - 6i Then (x^2 + y^2)^2 =(x + iy)^2 * (x - iy)^2 (9 + 6i)*(9 - 6i) = 117 Hence, x^2 + y^2 = (117)^(1/2) Finally, we have (x + y)^2 = x^2 + y^2 + 2xy (x + y)^2 = 6 + (117)^(1/2) So, x + y = +-[6 + (117)^(1/2)]^(1/2)

[@zabitkhalifakuliyev2271]

Hi.İ wanted to know if there is a formula for calculating the sum of a progression with a variable difference.I couldn't find it anywhere.So i decided to ask you.And if there is ,does this formula coincide with mine?Can you please answer? Sn=((2a1+(n-1)d1)×n)/2+(n(n-1)(n-2)×k)/6 d1-difference between first and second numbers How much does the difference changes each time is k

[@MathBooster]

In any non standard form of sequence, the best way to solve the problem is : Find the nth term of the sequence, and then find the sum of the sequence by using the nth term.

[@MathBooster]

Also there is a formula, if the difference of terms is in AP. In that case, nth term (Tn) = an² + bn + c Now put n=1, 2 and 3 to get 3 equations in a, b and c. And then solve the 3 equations. Also, if the difference of terms is in GP, then nth term (Tn) = a•r^(n-1) + bn + c. Now again you can find the value of a, b and c by putting n= 1, 2 and 3.

[@zabitkhalifakuliyev2271]

Thanks🥰

[@sinsn2147]

Here's a way that is similar to both ways introduced in the video, but much more easier to understand. (x^2-y^2)^2=(x+y)^2 *(x-y)^2=81 Let x+y=t t^2*(t^2-4*xy)=t^2*(t^2-4*3)=81 (t^2)^2-12*t^2-81=0 Then same as the last two minutes of the video.

[@guyhoghton399]

Method 1a: x⁴ + y⁴ = 99 as per method 1. Let u = x + y, v = x - y. u² = x² + y² + 2xy = x² + y² + 6. v² = x² + y² - 2xy = x² + y² - 6. ∴ u² - v² = 12 ... ➀ u² + v² = 2(x² + y²) ∴ (u² + v²)² = 4(x⁴ + y⁴ + 2(xy)²) = 4(99 + 18) = 4(117). ∴ u² + v² = 6√13 (u, v are real). Adding ➀ to the above: 2u² = 12 + 6√13 ∴ x + y = u = ±√(6 + 3√13).

[@lawrong1172]

Are you from Thailand?

[@ИгорьКулик-л2р]

Все решается проще и в уме

[@amnonnesher5253]

X=1, y=3 x=3 y=1

[@trakos-9448]

Amazing work

[@MathBooster]

Thank you 🙂

[@pentagon-math]

youtube.com/@pentagon-math

[@abdessamadsafri8791]

Between that 1 = -1

[@ناصرالعتيبي-ك1و3ب]

4.4

[@Mussxl]

4

[@JSSTyger]

I believe x+y = 4.100811

[@hanswust6972]

Nice trick!

[@pentagon-math]

youtube.com/@pentagon-math

[@dellagobaikal8205]

Can be resolved by composing a square equation in complex digits....😂

[@abdessamadsafri8791]

Olympia mat❤❤❤❤❤

[@idvbane8580]

V gd explain🥰

[@MathBooster]

Thanks 😊

[@eljano1728]

√117=3√13

[@JeanandreAlmeida-v4g]

×y=1

[@edgarkerr234]

9 is wrong. I’m sorry but your logic is incorrect. You changed the parenthesis to multiplication without performing the multiplication first. Algebraic distribution dictates that, in order to remove a parenthesis, we must multiply the factors individually before removing the parenthesis

[@abdessamadsafri8791]

Mat❤❤

[@idvbane8580]

Matg is ❤

[@idvbane8580]

Me 4🥰🥰

[@alinayfeh4961]

The correct Answer (x+y)= 6+√(117)

[@idvbane8580]

🥰🥰

[@idvbane8580]

🥰🥰good night

[@ИгорьКулик-л2р]

Сложно

[@idvbane8580]

Algebra🥰

[@pentagon-math]

youtube.com/@pentagon-math

[@idvbane8580]

Sir🥰🥰

[@probetting]

X=3 - Y=1 => X+Y = 4

[@mkngani4718]

??+ _/1+56 ? ÷57/ +( ) ×98 +( ஒரு நாள்) உள்ள மக்கள் மத்தியில் ஒரு நாள் ×70+ ஒரு நாள் இரவு நேரத்தில்....தமிழ் நாட்டில் இருக்கும் mks....

[@idvbane8580]

Matg 🥰

[@osadajayasinghe-cu5db]

Your 9 is like g 😂... but well done

[@sandeepsinghbawa5076]

4

[@kantilaljain180]

4

[@thegastronomicalhistorian9790]

4