√x+1 -√9-x= √2x-12

Let's analyze and solve the equation step by step: Given: \[ \sqrt{x + 1} - \sqrt{9 - x} = \sqrt{2x - 12} \] **Step 1: Determine the domain** - For \(\sqrt{x+1}\), we need \(x + 1 \geq 0\) \(\Rightarrow x \geq -1\). - For \(\sqrt{9 - x}\), we need \(9 - x \geq 0\) \(\Rightarrow x \leq 9\). - For \(\sqrt{2x - 12}\), we need \(2x - 12 \geq 0\) \(\Rightarrow x \geq 6\). Combining these: \[ x \geq 6 \quad \text{and} \quad x \leq 9 \] So, the domain is \([6, 9]\). --- **Step 2: Isolate and manipulate the equation** Start with: \[ \sqrt{x + 1} - \sqrt{9 - x} = \sqrt{2x - 12} \] Let's move all square roots to one side: \[ \sqrt{x + 1} = \sqrt{9 - x} + \sqrt{2x - 12} \] --- **Step 3: Square both sides to eliminate the square root** \[ (\sqrt{x + 1})^2 = (\sqrt{9 - x} + \sqrt{2x - 12})^2 \] Calculate: \[ x + 1 = (9 - x) + (2x - 12) + 2 \sqrt{(9 - x)(2x - 12)} \] Simplify: \[ x + 1 = 9 - x + 2x - 12 + 2 \sqrt{(9 - x)(2x - 12)} \] \[ x + 1 = (9 - 12) + (-x + 2x) + 2 \sqrt{(9 - x)(2x - 12)} \] \[ x + 1 = -3 + x + 2 \sqrt{(9 - x)(2x - 12)} \] Subtract \(x\) from both sides: \[ 1 = -3 + 2 \sqrt{(9 - x)(2x - 12)} \] Add 3 to both sides: \[ 4 = 2 \sqrt{(9 - x)(2x - 12)} \] Divide both sides by 2: \[ 2 = \sqrt{(9 - x)(2x - 12)} \] Now, square both sides again: \[ (2)^2 = (9 - x)(2x - 12) \] Calculate: \[ 4 = (9 - x)(2x - 12) \] --- **Step 4: Expand and form a quadratic** Expand the right side: \[ 4 = (9)(2x - 12) - x(2x - 12) \] \[ 4 = 18x - 108 - 2x^2 + 12x \] Simplify: \[ 4 = (18x + 12x) - 108 - 2x^2 \] \[ 4 = 30x - 108 - 2x^2 \] Bring all to one side: \[ 2x^2 - 30x + 112 = 0 \] Divide through by 2: \[ x^2 - 15x + 56 = 0 \] --- **Step 5: Solve the quadratic** Factors of 56 that sum to 15: \[ x^2 - 15x + 56 = 0 \] Factor: \[ (x - 7)(x - 8) = 0 \] Solutions: \[ x = 7 \quad \text{or} \quad x = 8 \] Recall the domain: \([6, 9]\). Both are within the domain. --- **Step 6: Verify solutions in the original equation** **Check \(x=7\):** Left side: \[ \sqrt{7 + 1} - \sqrt{9 - 7} = \sqrt{8} - \sqrt{2} \approx 2.828 - 1.414 = 1.414 \] Right side: \[ \sqrt{2 \times 7 - 12} = \sqrt{14 - 12} = \sqrt{2} \approx 1.414 \] Equal, so **\(x=7\)** is a solution. --- **Check \(x=8\):** Left side: \[ \sqrt{8 + 1} - \sqrt{9 - 8} = \sqrt{9} - \sqrt{1} = 3 - 1 = 2 \] Right side: \[ \sqrt{2 \times 8 - 12} = \sqrt{16 - 12} = \sqrt{4} = 2 \] Equal, so **\(x=8\)** is a solution. --- ### **Final answer:** \[ \boxed{ \text{Solutions are } x=7 \quad \text{and} \quad x=8 } \] both within the valid domain and satisfying the original equation.