Questions: Solve for (x), where (x) is a real number. [ sqrt7 x+6-sqrt5 x+16=0 ] If there is more than one solution, separate them with commas. If there is no solution, click on "No solution". [ x= ]

Solution Steps

To solve the given radical equation, we need to isolate one of the square roots and then square both sides to eliminate the radicals. This will simplify the equation to a linear form, which can then be solved for \( x \). After finding the potential solutions, we must check each one in the original equation to ensure they are valid, as squaring can introduce extraneous solutions.

We start with the equation: \[ \sqrt{7x + 6} - \sqrt{5x + 16} = 0 \] To isolate one of the square roots, we rearrange the equation: \[ \sqrt{7x + 6} = \sqrt{5x + 16} \]

Next, we square both sides to eliminate the square roots: \[ 7x + 6 = 5x + 16 \]

Now, we simplify the equation: \[ 7x - 5x = 16 - 6 \] This simplifies to: \[ 2x = 10 \] Dividing both sides by 2 gives: \[ x = 5 \]

We substitute \( x = 5 \) back into the original equation to verify: \[ \sqrt{7(5) + 6} - \sqrt{5(5) + 16} = \sqrt{35 + 6} - \sqrt{25 + 16} = \sqrt{41} - \sqrt{41} = 0 \] Since the left-hand side equals zero, \( x = 5 \) is a valid solution.

Final Answer