To solve the equation √6x+3 = x-2, we can start by squaring both sides of the equation:
(√6x+3)² = (x-2)²
Simplifying the left side of the equation, we get:
6x+3 = (x-2)²
Expanding the right side of the equation, we get:
6x+3 = x² - 4x + 4
Moving all the terms to one side, we get a quadratic equation:
x² - 10x + 1 = 0
To solve this quadratic equation, we can use the quadratic formula:
x = (-b ± √(b² - 4ac)) / 2a
Where a = 1, b = -10, and c = 1. Plugging these values into the formula, we get:
x = (10 ± √(100 - 4))/2
x = (10 ± √96)/2
x = 5 ± 2√6
Therefore, the solutions to the equation √6x+3 = x-2 are x = 5 + 2√6 and x = 5 - 2√6.