Video

It took me a few seconds to see it was 1/2.

Nice!

Cross multiplication also gives 4x^2=1 meaning x^2=1/4 and so x=1/2. A much l;ess complicated solution than yours.

I got it in 15 seconds through vedic mental calculations Edit. Vedic maths is ancient system of calculation in ancient India

We can frame a quadratic equation in t^2+ 6t+ 31 =0

Great explanation.

2^x = 5, then you put in log base 2 of x is 5

Die Argumentation mit den Graphen und Ihrer Symmetrie ist doch viel einfacher! Und es lässt sich zeigen, dass es genau einen Schnittpunkt gibt.

Nice one!

This is an very easy question, i don't believe it's a Harvard interview question.

by faktoring , x^3-21x^2 +21x^2 -441x +141x-2961=0 , (x-21)(x^2+21x+141)=0 , x-21=0 , x=21 , test , 21^3-300*21=9261-6300 , 9261-6300=2961 , for complex , x^2+21x+141=0 , solu., x = 21, (-21+i*sqrt123)/2 , (-21-i*sqrt123)/2 ,

If you don't spot a solution (x=5) the cubic can easily be solved by radicals.

a

The answers found (6,5) (-5,-6) and (4,-3) (3,-4) are the difference and sum of cubes respectively. Initially I visualized the difference of 2 cubes before watching. The difference of the cube sides, delta, gives a volume difference between the cubes of 91. Taking the smaller cube side, a, (absolute), the size of the volume difference is 3 x a^2 x delta + 3 x a x delta^2 + delta^3 = 91. In order to do the problem mentally I assumed delta was a whole number. Considering a^2 x delta < 30, and the smallest whole number for delta is 1, the largest candidate a is 5, giving a solution of (6,5). (I didn't consider the sum of cubes. ) Once the integer cubes cases are found, the equation for all real solutions has example test cases.

Even before I started to think, my intuition said me to check negatives. Then, it is a sum of square and third power. 4+8, so, the answer is -2. This is the solution if the problem is supposed for 12-years old children. For Oxford, you also need the complex ones

Isnt this a polynomial and so shouldn't it have 4 roots

It's easier to replace a by x + ½. Then the powers to the 4 reduce immediately to 2x³ + ½x = 0

Is this video a wind up? It’s how not to do a maths question!

2^x + 8^x = 130 2^x + 4 (8^x) = 130 2^x (1+4) = 130 2^x × 5 = 130 2^x = 130 ÷ 5 2^x = 26 xln(2) = ln(26) __________________ |x = ln(26)/ln(2)| __________________ |x ≈ 4.70043... |

Olympiade? 😂😂😂 For 3 years old kids?

At a quick glance I start by taking the square root of both sides. Then a^2 = a^2 -2a +1.Then 2a -1 = 0 and -2a + 1 = 0. Then a = + 1/2 and - 1/2

Yes, or ... ˣ√2 = 3 2 = 3ˣ note: log₃(2) = 0.630930 3⁰·⁶³⁰⁹³⁰ = 3ˣ same base (= 3) ■ x = 0.630930 🙂

Taking the square root of a negative number seems weird (the first reason is that (-i)^2 = i^2 = -1, so sqrt(-1) could be i or -i...). I have never seen that this is allowed, but maybe I don't know this formalism.

After the root 21 is found, synthetic division avoids the rest of the slog. The observation any mathlete would make is the sum of the roots is zero. This would also lead to the analysis.

a=2021, b=0, c=2021

would it be cheap to pull out a simplified cubic formula?

MPRAVISSIMO!!!

2^x=5

A way to avoid the division: a generic expansion of a cubic (with roots a,b,c) gives constant abc=2961 or ab=141 when you sub in 21. the x^2 coef is a+b+c=0 ie. a+b+21=0, you can sub in b=141/a, simplify, and you have a familiar quadratic to solve for the complex roots.

Try a=(a-1), -(a-1), i(a-1) , -i(a-1) a=1/2, (1-i)/2, (1+i)/2

my direct impulse: 1. a1 = 0.5 2. a^4 cancels out -> polynomial 3rd order 3. polynomial division by (x-0.5) 4. abc (or pq) formula (I'm German)

❤ល្អណាស់🇰🇭🇰🇭🇰🇭🇰🇭🙏🙏🙏good

Or we could just write it as: One hundred and dirty and solve it for y.

🇰🇭🇰🇭🇰🇭👍👍👍ល្អណាស់good

a = 0.5?

OK Space Cadets ..this should be easy 2^x + 4^x + 8^x + 16^x = 780 ........................ hint: the 3^x + 9^x + 27^x + 81^x = 780 results in x = Log to the base 3 of 5 :-)

I think that you are making these up as you go along. Is this realy from Harvard, was the other one realy from Cambridge? They are good fun but perhaps not as hard as your titles suggest?

Log base 2 of 5 falls out by simple inspection! Why so difficult?

I can't see anything wrong with: a = 1/2 a⁴ = 0.0625000 (a - 1)⁴ = 0.0625000 Is it an exercise of problem solving or problem making?

It's a 8th grade problem 😅

33% the question said nothing about “real solutions” 😜

As soon as I saw that we had a cubic to solve, and 130 has a prime factorisation to 3 primes, 2x5x13, I knew one of them was going to be a candidate! Also if you dont do the 26-25 trick, and consider what the coeficients of general cubic would be, given there is no y^2 coeficient, you get the property that the sum of the roots is 0, and the product of the roots is 26.

Far easier approach, factor both sides: (1) LHS: x²-x³ = x²•(1-x) (2) RHS: 12= 4•3 = 2²•3 From (1) & (2) we have x²=2² and (x-1)=3. Solving the second equation we get x=-2.

Just guess that the answer must be a bit bigger than 2 and then iterate. Much easier and faster than his baloney method

Awesome. I'll show this to my Year 2 class tomorrow

you are missing a bunch of solutions. there are two more real solutions at -0.81881 and 1.41138, and then there are 3 complex solutions also.

To be an Oxford university exam is not so difficult, actually with a bit of creativity I' d dare to say that is quite easy 😂

Cadê a quarta solução?

There must be 4 solutions !

I got the solutions of Lambert W function. idk