Solve for \( u \), where \( u \) is a real number. \[ \sqrt{34-5 u}=u-4 \] If there is more than one solution, separate them with commas. If there is no solution, click on "No solution". \( u=\square \square \)

To solve the equation \( \sqrt{34 - 5u} = u - 4 \), we begin by squaring both sides to eliminate the square root: \[ 34 - 5u = (u - 4)^2 \] Expanding the right side gives: \[ 34 - 5u = u^2 - 8u + 16 \] Rearranging this equation leads us to: \[ 0 = u^2 - 3u - 18 \] This is a quadratic equation, and we can factor it: \[ 0 = (u - 6)(u + 3) \] Setting each factor to zero gives us the potential solutions: \[ u - 6 = 0 \quad \Rightarrow \quad u = 6 \] \[ u + 3 = 0 \quad \Rightarrow \quad u = -3 \] Now we need to check these solutions in the original equation to ensure they don't introduce extraneous results. 1. **For** \( u = 6 \): \[ \sqrt{34 - 5(6)} = \sqrt{34 - 30} = \sqrt{4} = 2 \] And on the right side: \[ 6 - 4 = 2 \] This satisfies the equation. 2. **For** \( u = -3 \): \[ \sqrt{34 - 5(-3)} = \sqrt{34 + 15} = \sqrt{49} = 7 \] And on the right side: \[ -3 - 4 = -7 \] This does not satisfy the equation. Thus, the only valid solution is: \[ u = 6 \] So, the final answer is: \( u=6 \)