Seemingly Impossible Algebra Question

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Seemingly Impossible Algebra Question
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21 окт 2023

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Комментарии : 22

@NadiehFan 9 месяцев назад

Solving the quadratic equation in x at the end is redundant. From the system a + b = 5 a⁴ + b⁴ = 97 we can derive that we have either ab = 6 or ab = 44. From a + b = 5, ab = 6 we get the solutions (a, b) = (2, 3) and (a, b) = (3, 2) and since x = b⁴ it is immediately clear that this gives either x = 2⁴ = 16 or x = 3⁴ = 81.

@guyhoghton399 9 месяцев назад

When you get the value of _ab_ it is simpler to complete the solution using Vieta's formula: _a_ and _b_ are the roots of _y² - (a + b)y + ab = 0_ For the case _ab = 6_ : _y² - 5y + 6 = 0_ ⇒ _(y - 2)(y - 3) = 0_ ∴ _b = 2_ or _3_ ∴ _x = b⁴ = 16_ or _81_

@luigid6205 9 месяцев назад

Since I'm not that much of an expert I've ended up taking a different path (which I guess works only with x being both a real number and an integer). I started solving by limiting the range of x's possible values with the existence condition (0

@idvbane8580 9 месяцев назад

🥰🥰

@honestadministrator 9 месяцев назад

RHS being an integer both x & 97- x must be 4 th power of integers 4 th power of integers less than or equal to 97 are 81, 16 and 1 . Interestingly 81 + 16 = 97 Hereby x = 16, 81

@robertveith6383 9 месяцев назад

How would you know for sure at the beginning. Maybe (97 - x) and x would be equal to fractions with denominators of 16. Then, their fourth roots would have denominators of 2. Finally, the sum would equal 5.

@carly09et 9 месяцев назад

@@robertveith6383 Guess and check give x = 16 or 81 that there is only two solutions needs number theory that the other 'solutions' are complex. 97-x === {2^4,3^4} & x === {3^4,2^4} 97 == a^4 + b^4 & a + b == 5

@misterenter-iz7rz 9 месяцев назад

Simply 16.

@padraiggluck2980 9 месяцев назад

Literally 10 secs.

@user-er9fx3qx7n 9 месяцев назад

Right 😊

@dmwallacenz 9 месяцев назад

Yeah, finding two possible values for x in 10 seconds is easy. Proving they're the only two is much harder. Unless you've done the latter, you haven't solved the equation. So no, I don't believe you solved this in 10 seconds.

@padraiggluck2980 9 месяцев назад

@@dmwallacenz nonsense

@dmwallacenz 9 месяцев назад

Really? So how did you prove in 10 seconds that 16 and 81 are the only solutions? You must be aware of some advanced mathematical techniques that nobody else in the world knows about.

@padraiggluck2980 9 месяцев назад

@@dmwallacenz It takes longer to explain than to do mentally. 4 is a special number. Fourth roots call for fourth powers and they cannot be large so start with 2^4. Note that i^4 =1, also. So subtract 16 from 95 and see if we get a fourth power. 81 is 3^4. So the solution set is {+-2, +- 2i} . Any middle school child in China, Japan, India, and many other countries can just write out the answer to this problem immediately. Provide a counterexample that shows that the solution is not unique.

@yakupbuyankara5903 7 месяцев назад

X=81

@nasrullahhusnan2289 9 месяцев назад

x=81

@prabhagupta6871 9 месяцев назад

Not that much difficult as compared to other questions on your channel

@walterwen2975 9 месяцев назад

Seemingly Impossible Algebra Question: ⁴√(97 - x) + ⁴√x = 5, Find all real values of x ⁴√(97 - x) + ⁴√x = 5; 5 > ⁴√(97 - x) > 0, 5 > ⁴√x > 0, 97 > x > 0 First method: ⁴√(97 - x) + ⁴√x = 5 = 3 + 2 = ⁴√(3⁴) + ⁴√(2⁴) = ⁴√81 + ⁴√16 = ⁴√(97 - 16) + ⁴√16 = ⁴√16 + ⁴√81 = ⁴√(97 - 81) + ⁴√81; x = 16 or x = 81 Second method: Let: y = ⁴√(97 - x), z = ⁴√x; y⁴ + z⁴ = 97, yz = ⁴√[(97 - x)x], 97 > x > 0 (y + z)² = y² + z² + 2yz = 5², y² + z² = 25 - 2yz, (y² + z²)² = y⁴ + z⁴ + 2(yz)² = (25 - 2yz)² 97 + 2(yz)² = 25² - 100yz + 4(yz)², (yz)² - 50yz + 264 = 0, (yz - 6)(yz - 44) = 0 yz - 6 = 0; yz = 6 or yz - 44 = 0; yz = 44 yz = ⁴√(97x - x²) = 6, 97x - x² = 6⁴ = 1296 or yz = ⁴√(97x - x²) = 44, 97x - x² = 44⁴ x² - 97x + 1296 = 0 or x² - 97x + 44⁴ = 0; Rejected, complex roots without real value (x - 16)(x - 81) = 0, x - 16 = 0; x = 16 or x - 81 = 0; x = 81 Answer check: x = 16 or x = 81 ⁴√(97 - x) + ⁴√x = 5; Confirmed in First method Final answer: x = 16 or x = 81