Questions: Solve the following radical equation. If needed, write your answer as a fraction reduced to lowest terms. sqrt(9-6x) - sqrt(7x+3) = 0 x =

Solution Steps

To solve the given radical equation, we need to isolate one of the square root terms and then square both sides to eliminate the square roots. This process may need to be repeated if there are multiple square roots. After eliminating the square roots, solve the resulting equation for \( x \). Finally, check the solutions in the original equation to ensure they do not produce any extraneous solutions.

Starting with the equation

\[ \sqrt{9 - 6x} - \sqrt{7x + 3} = 0, \]

we isolate one of the square roots:

\[ \sqrt{9 - 6x} = \sqrt{7x + 3}. \]

Next, we square both sides to eliminate the square roots:

\[ 9 - 6x = 7x + 3. \]

Rearranging the equation gives:

\[ 9 - 3 = 7x + 6x, \]

which simplifies to:

\[ 6 = 13x. \]

Dividing both sides by 13, we find:

\[ x = \frac{6}{13}. \]

We substitute \( x = \frac{6}{13} \) back into the original equation to verify:

\[ \sqrt{9 - 6\left(\frac{6}{13}\right)} - \sqrt{7\left(\frac{6}{13}\right) + 3} = 0. \]

Calculating each term:

1. \( 9 - 6\left(\frac{6}{13}\right) = 9 - \frac{36}{13} = \frac{117 - 36}{13} = \frac{81}{13} \)
2. \( 7\left(\frac{6}{13}\right) + 3 = \frac{42}{13} + \frac{39}{13} = \frac{81}{13} \)

Thus, we have:

\[ \sqrt{\frac{81}{13}} - \sqrt{\frac{81}{13}} = 0, \]

which confirms that \( x = \frac{6}{13} \) is a valid solution.

Final Answer