To solve this equation, we need to first expand and simplify both sides:
((x-3)² + (x-4)²) - (x-5)² - x = 24(x² - 6x + 9 + x² - 8x + 16) - (x² - 10x + 25) - x = 24(2x² - 14x + 25) - x² + 10x - 25 - x = 242x² - 14x + 25 - x² + 10x - 25 - x = 24x² - 5x = 24
Next, we can move all terms to one side to set the equation equal to zero:
x² - 5x - 24 = 0
Finally, we can factor the quadratic equation:
(x - 8)(x + 3) = 0
Setting each factor to zero gives us two possible solutions:
x - 8 = 0x = 8
x + 3 = 0x = -3
Therefore, the solutions to the equation are x = 8 and x = -3.
To solve this equation, we need to first expand and simplify both sides:
((x-3)² + (x-4)²) - (x-5)² - x = 24
(x² - 6x + 9 + x² - 8x + 16) - (x² - 10x + 25) - x = 24
(2x² - 14x + 25) - x² + 10x - 25 - x = 24
2x² - 14x + 25 - x² + 10x - 25 - x = 24
x² - 5x = 24
Next, we can move all terms to one side to set the equation equal to zero:
x² - 5x - 24 = 0
Finally, we can factor the quadratic equation:
(x - 8)(x + 3) = 0
Setting each factor to zero gives us two possible solutions:
x - 8 = 0
x = 8
x + 3 = 0
x = -3
Therefore, the solutions to the equation are x = 8 and x = -3.