2) \( 3 \sqrt{x+4}=x \)

To solve the equation \( 3 \sqrt{x+4}=x \), first, isolate the square root by dividing both sides by 3, giving you \( \sqrt{x+4} = \frac{x}{3} \). Next, square both sides to eliminate the square root: \( x+4 = \frac{x^2}{9} \). Multiply the whole equation by 9 to clear the fraction: \( 9x + 36 = x^2 \). Rearranging gives \( x^2 - 9x - 36 = 0 \). Use the quadratic formula \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \) to find the values of \( x \). Real-world applications of quadratic equations like this one can be found in various fields, such as engineering and physics, where understanding the shape of a trajectory or optimizing designs is crucial. Solving such equations often reveals important insights about positions, forces, or materials' properties, making them not just theoretical exercises but practical tools for innovation. A common mistake when dealing with equations involving square roots is to forget to check for extraneous solutions. After squaring both sides, there's a risk of introducing solutions that don't satisfy the original equation. So after solving \( x^2 - 9x - 36 = 0 \), be sure to substitute the potential solutions back into the original equation \( 3 \sqrt{x+4} = x \) to ensure they are valid!