\( \begin{array}{llll}\text { 9. } \sqrt{2 x+22}-7=x & \text { b. } 9 \text { and }-3 & \text { c. }-3 & \text { d. }-9 \text { and }-3 \\ \text { a. }-9 & \end{array} \)

Let's start with the equation \( \sqrt{2x + 22} - 7 = x \). To solve for \( x \), first, isolate the square root: \[ \sqrt{2x + 22} = x + 7 \] Next, square both sides to eliminate the square root: \[ 2x + 22 = (x + 7)^2 \] Expanding the right side, we have: \[ 2x + 22 = x^2 + 14x + 49 \] Rearranging gives us a quadratic equation: \[ 0 = x^2 + 12x + 27 \] Now we can factor the quadratic: \[ 0 = (x + 9)(x + 3) \] Setting each factor to zero gives us the solutions: \[ x + 9 = 0 \quad \Rightarrow \quad x = -9 \] \[ x + 3 = 0 \quad \Rightarrow \quad x = -3 \] Checking for extraneous solutions, both values satisfy the original equation. Therefore, the solutions are \( x = -9 \) and \( x = -3 \). In the real world, solving quadratic equations like this is crucial in fields such as engineering, finance, and physics, where you might need to determine values like projectile heights, investment returns, or structural loads. Understanding these principles helps in developing critical thinking skills that can be applied to solve everyday problems. If you're tackling equations, remember that squaring both sides can introduce extraneous solutions—double-checking your answers by plugging them back into the original equation helps keep your solutions valid and accurate! It's a common pitfall; always verify!