Solve the following quadratic equation for all values of $x$ in simplest form.
\[
3(3 x+6)^{2}-46=5
\]

Answer

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Answer

Thus, the solutions to the equation are $x = \boxed{-2 - \frac{\sqrt{17}}{3}}$ and $x = \boxed{-2 + \frac{\sqrt{17}}{3}}$.

Steps

Step 1 ：Given the quadratic equation $3(3x+6)^{2}-46=5$.

Step 2 ：Simplify the equation to get $3(3x+6)^{2}=51$.

Step 3 ：Divide both sides by 3 to get $(3x+6)^{2}=17$.

Step 4 ：Take the square root of both sides to get $3x+6=\pm\sqrt{17}$.

Step 5 ：Subtract 6 from both sides to get $3x=\pm\sqrt{17}-6$.

Step 6 ：Divide both sides by 3 to get $x=\pm\frac{\sqrt{17}-6}{3}$.

Step 7 ：Thus, the solutions to the equation are $x = \boxed{-2 - \frac{\sqrt{17}}{3}}$ and $x = \boxed{-2 + \frac{\sqrt{17}}{3}}$.