Solve the equation: \( \sqrt{x+24}=x+4 \) Answer: \( x=\square \) Question Help: \( \square \) Message instructor

To solve the equation \( \sqrt{x+24}=x+4 \), we start by squaring both sides to eliminate the square root: \[ x + 24 = (x + 4)^2 \] Expanding the right side gives: \[ x + 24 = x^2 + 8x + 16 \] Now, let's rearrange the equation to bring everything to one side: \[ 0 = x^2 + 8x + 16 - x - 24 \] This simplifies to: \[ 0 = x^2 + 7x - 8 \] Next, we can factor this quadratic equation: \[ (x + 8)(x - 1) = 0 \] Setting each factor to zero gives us the potential solutions: \[ x + 8 = 0 \quad \Rightarrow \quad x = -8 \] \[ x - 1 = 0 \quad \Rightarrow \quad x = 1 \] Now, we need to check if these values satisfy the original equation since we squared it, which could introduce extraneous solutions. For \( x = -8 \): \[ \sqrt{-8 + 24} = \sqrt{16} = 4 \quad \text{and} \quad -8 + 4 = -4 \quad \text{(not a solution)} \] For \( x = 1 \): \[ \sqrt{1 + 24} = \sqrt{25} = 5 \quad \text{and} \quad 1 + 4 = 5 \quad \text{(this works)} \] Therefore, the only solution is: \[ x = 1 \] Answer: \( x=1 \)