Solving a radical equation with a quadratic expression under the radical Solve for \( x \), where \( x \) is a real number. \( x=\sqrt{x^{2}+x+6} \) If there is more than one solution, separate them with commas. If there is no solution, click on "No solution".

To solve the equation: \[ x = \sqrt{x^{2} + x + 6} \] **Step 1: Square Both Sides** To eliminate the square root, square both sides of the equation: \[ x^2 = x^2 + x + 6 \] **Step 2: Simplify the Equation** Subtract \( x^2 \) from both sides: \[ 0 = x + 6 \] **Step 3: Solve for \( x \)** \[ x = -6 \] **Step 4: Verify the Solution** It's important to check whether this solution satisfies the original equation: \[ -6 = \sqrt{(-6)^2 + (-6) + 6} \] \[ -6 = \sqrt{36 - 6 + 6} \] \[ -6 = \sqrt{36} \] \[ -6 = 6 \] This is not true. Therefore, \( x = -6 \) is an **extraneous solution** and does not satisfy the original equation. **Conclusion:** There is **no real solution** to the equation. **Answer:** No solution