6:45 5G Solving Radical Equations Radical Functions Solve the equation. \[ \begin{array}{c} \sqrt{14 x-6}-3=x \\ x=[?], \square \end{array} \] Enter the smallest solution first. Back Reload Home Profile Menu

To solve the equation \(\sqrt{14x - 6} - 3 = x\), first, isolate the square root: \[\sqrt{14x - 6} = x + 3.\] Next, square both sides to eliminate the square root: \[14x - 6 = (x + 3)^2.\] Expanding the right side gives: \[14x - 6 = x^2 + 6x + 9.\] Rearranging the equation leads to: \[x^2 - 8x + 15 = 0.\] Factoring gives: \[(x - 3)(x - 5) = 0.\] Thus, the solutions are \(x = 3\) and \(x = 5\). The smallest solution is \(x = 3\). So, the final answer is: \[x = 3, \square x = 5.\] In real-world applications, radical equations can often arise in physics when calculating distances, speeds, or even in engineering when dealing with structural loads. It's vital to understand how to manipulate these equations, as solving them correctly can lead to practical solutions in various fields. Also, when solving radical equations, a common mistake is neglecting to check for extraneous solutions. Squaring both sides can introduce solutions that don't satisfy the original equation. After finding the solutions, always substitute them back into the original equation to ensure they are valid!