To solve this equation, we will first expand the left side:
(2x-1)³ = (2x-1)(2x-1)(2x-1= (4x² - 2x - 2x + 1)(2x-1= (4x² - 4x + 1)(2x-1= 8x³ - 8x² + 2x² - 2= 8x³ - 6x² - 2x
Now, substitute this expression back into the original equation:
8x³ - 6x² - 2x - 4x²(2x-3) = 18x³ - 6x² - 2x - 8x² + 12x = 18x³ - 14x² + 10x - 17 = 0
This is a cubic equation which is difficult to solve algebraically. One common technique to solve a cubic equation is to use the Rational Root Theorem to try potential rational roots and find one that makes the equation equal to zero.
To solve this equation, we will first expand the left side:
(2x-1)³ = (2x-1)(2x-1)(2x-1
= (4x² - 2x - 2x + 1)(2x-1
= (4x² - 4x + 1)(2x-1
= 8x³ - 8x² + 2x² - 2
= 8x³ - 6x² - 2x
Now, substitute this expression back into the original equation:
8x³ - 6x² - 2x - 4x²(2x-3) = 1
8x³ - 6x² - 2x - 8x² + 12x = 1
8x³ - 14x² + 10x - 17 = 0
This is a cubic equation which is difficult to solve algebraically. One common technique to solve a cubic equation is to use the Rational Root Theorem to try potential rational roots and find one that makes the equation equal to zero.