Final answer:
To solve the radical equation √(8x+9) = x+2, square both sides, rearrange into a quadratic equation, factor, find the solutions, and check for extraneous solutions. Both x = 5 and x = -1 are valid solutions and there are no extraneous solutions.
Step-by-step explanation:
To solve the radical equation √(8x+9) = x+2, first square both sides to eliminate the square root, resulting in 8x + 9 = (x + 2)². Expanding the right side gives 8x + 9 = x² + 4x + 4. Rearrange this equation to form a quadratic equation: x² - 4x - 5 = 0. Factor the quadratic to (x - 5)(x + 1) = 0, yielding solutions x = 5 and x = -1.
However, these solutions must be checked in the original equation to ensure they are not extraneous. Inserting x = 5 gives √(40+9) = 5 + 2 which is true, whereas inserting x = -1 gives √(-8+9) = -1 + 2, which is also true; therefore, there are no extraneous solutions for this equation.