Bot GPT 4-o mini To solve the equation
\[ \sqrt{2x + 5} - 2 = \sqrt{2x - 3}, \]
we start by isolating one of the square root terms. We can do this by adding 2 to both sides:
\[ \sqrt{2x + 5} = \sqrt{2x - 3} + 2. \]
Next, we square both sides to eliminate the square roots:
\[ (\sqrt{2x + 5})^2 = (\sqrt{2x - 3} + 2)^2. \]
This simplifies to:
\[ 2x + 5 = (2x - 3) + 4\sqrt{2x - 3} + 4. \]
Combining like terms on the right side gives us:
\[ 2x + 5 = 2x + 1 + 4\sqrt{2x - 3}. \]
Now we can simplify the equation:
\[ 5 = 1 + 4\sqrt{2x - 3}. \]
Subtract 1 from both sides:
\[ 4 = 4\sqrt{2x - 3}. \]
Dividing both sides by 4 results in:
\[ 1 = \sqrt{2x - 3}. \]
Next, we square both sides again to eliminate the square root:
\[ 1^2 = (2x - 3) \implies 1 = 2x - 3. \]
Adding 3 to both sides gives us:
\[ 4 = 2x. \]
Dividing by 2 results in:
\[ x = 2. \]
Finally, it’s important to check if this solution satisfies the original equation. We substitute \(x = 2\) back into the original equation:
\[ \sqrt{2(2) + 5} - 2 = \sqrt{2(2) - 3}. \]
Calculating both sides:
\[ \sqrt{4 + 5} - 2 = \sqrt{4 - 3} \implies \sqrt{9} - 2 = \sqrt{1} \implies 3 - 2 = 1. \]
Both sides equal 1, meaning our solution is verified.
Thus, the solution to the equation is
\[ \boxed{2}. \]
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