A Very Nice Exponential Equation | Math Olympiad Preparation

A Very Nice Exponential Equation | Math Olympiad Preparation
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From basic concepts to advanced problem-solving techniques, we cover everything you need to know to excel in Olympiad-level math competitions. Join us as we unravel the complexities of exponential equations and provide valuable insights and strategies to tackle them with confidence. Whether you're a seasoned competitor or just getting started, this video will sharpen your skills and elevate your performance to new heights!
Topics covered:
Exponential equations
How to solve exponential equations
Algebra
Algebraic identities
Synthetic division
Rational root theorem
Exponential Equation
Math Olympiad preparation
Math Olympiad training
Exponent laws
Solving cubic equation
Quadratic equation
Factorization
Real solutions
Timestamps:
0:00 Introduction
0:30 Exponent laws
2:30 Substitution
5:35 Solving cubic equation
7:45 Synthetic division
8:30 Quadratic equation
9:02 Factorization
10:45 Solutions
10:50 Verification
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Thanks for Watching !!
@infyGyan

[@user-ee7nw2rx9s]

Красиво

[@dorkmania]

First expressing 4^x - 4 as (2^x)² -2² and using difference of squares.
Then 12 as (4)(3), factoring the 3 outside the bracket as 3³ and multiplying both sides of the equation with 3³.
(((2^x - 2)(2^x + 2))/4)³ = ((2^x - 2)/2)⁴
Rearranging (2^x + 2) as (2^x - 2 + 4), 4 as (2)(2), then substituting ((2^x - 2)/2) = a
=> (a(a + 2))³ = 3³a⁴
=> a³(a + 2)³ - 27a⁴ = 0
=> a³(a³ + 6a² + 12a + 8 - 27a) = 0
=> a³(a³ + 6a² - 15a + 8) = 0
Using RRT and SDM with a = 1
=> a³(a - 1)(a² + 7a - 8) = 0
Factoring and simplifying
=> a³(a - 1)²(a+8) = 0
a = 0, 1, -8
Substituting back for x and simplifying
2^x = 2¹, 2², -14 (reject)
=> x = 1, 2

[@abcekkdo3749]

X=1,2

[@mohammedsaysrashid3587]

Another wonderful explanation... Thanks for sharing, Sir 🙏....t=2^(×-1) => x=1.x=2

[@infyGyan]

👍
Thanks for watching

[@Chacal0152]

t = 2^(x-1) => x = 1 or 2

[@user-nd7th3hy4l]

X=1 et certainement une autre.

[@ashokdubey8415]

using substitution t=2^x the given equation transforms to
(t^2-4)^3=108(t-2)^4
=> t=2,4
=>x=1,2

[@user-ny6jf9is3t]

×=1 ή ×=2

[@gregevgeni1864]

W=2^x, x=1 and x=2(double),
OR, no and .. I apologise 😊