A Nice Algebra Problem | Math Olympiad | How to solve?

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Find the value of x?
How to solve (x-3)^4 = 7^4

In this video, we'll show you How to Solve Math Olympiad Question A Nice Algebra Problem (x-3)^4 = 7^4 in a clear , fast and easy way. Whether you are a student learning basics or a professtional looking to improve your skills, this video is for you. By the end of this video, you'll have a solid understanding of how to solve math olympiad exponential equations and be able to apply these skills to a variety of problems.
#matholympiad #maths #math #algebra

[brianwade]

You went way off the rails at 3:03. There is a solution much simpler than what you did. You should have factored [(x-3)^2 - 7^2] as difference of two squares: (x-3+7)(x-3-7). This would have immediately given you roots {-4, 10}. Similarly you should have rewritten [(x-3)^2 + 7^2] as [(x-3)^2 - (-7^2))], also a difference of two squares, and then factored it as (x-3+7i)(x-3-7i). This would have immediately given you roots {3-7i, 3+7i}.

[key_board_x]

@ 2:40 / 10:09

(x - 3)² - 7² = 0
(x - 3)² = 7²
x - 3 = ± 7
x = 3 ± 7
→ x = 10
→ x = - 4


(x - 3)² + 7² = 0
(x - 3)² = - 7²
x - 3 = 7².i²
x = 3 ± 7i
→ x = 3 + 7i
→ x = 3 - 7i

[stpat]

I think my way is a bit easier:

(x - 3)^4 = 7^4
(x - 3)^(2 * 2) = 7^(2 * 2)
([x - 3]^2)^2 = (7^2)^2

Let a = (x - 3)^2, and b = 7^2
([x - 3]^2)^2 = (7^2)^2
=> a^2 = b^2
=> a^2 - b^2 = b^2 - b^2
=> a^2 - b^2 = 0
=> (a - b)(a + b) = 0
=> ([x - 3]^2 - 7^2)([x - 3]^2 + 7^2) = 0

Let c = x - 3, and d = 7
([x - 3]^2 - 7^2)([x - 3]^2 + 7^2) = 0
=> (c^2 - d^2)(c^2 + d^2) = 0
=> (c - d)(c + d)(c - d * i)(c + d * i) = 0
=> ([x - 3] - 7)([x - 3] + 7)([x - 3] - 7 * i)([x - 3] + 7 * i) = 0
=> (x - 3 - 7)(x - 3 + 7)(x - 3 - 7 * i)(x - 3 + 7 * i) = 0
=> (x - 10)(x + 4)(x - 3 - 7 * i)(x - 3 + 7 * i) = 0
=> x - 10 = 0, or x + 4 = 0, or x - 3 - 7 * i = 0, or x - 3 + 7 * i = 0

Suppose x - 10 = 0
x - 10 = 0
x - 10 + 10 = 0 + 10
x = 10

Suppose x + 4 = 0
x + 4 = 0
x + 4 - 4 = 0 = 4
x = -4

Suppose x - 3 - 7 * i = 0
x - 3 - 7 * i = 0
x - 3 - 7 * i + 3 + 7 * i = 0 + 3 + 7 * i
x = 3 + 7 * i

Suppose x - 3 + 7 * i = 0
x - 3 + 7 * i = 0
x - 3 + 7 * i + 3 - 7 * i = 0 + 3 - 7 * i
x = 3 - 7 * i

x1 = 10
x2 = -4
x3 = 3 + 7 * i
x4 = 3 - 7 * i

[فیروزاهنگری]

(x-3)^2=+-√7^4=+-7^2 result
(x-3)^2=7^2 or (x-3)^2=-7^2
case 1 (x-3)^2=7^2 then x-3=+-7
x=3+-7 x1=3+7=10 x2=3-7=-4
case2 (x-3)^2=-7^2 then x-3=+-√(-7^2) then x-3=+-7i
x=3+-7i
x3=3+7i x4=3-7i