Solve the equation
Does the equation have a solution?

To solve the equation , we first simplify the left side. Notice that can be factored as :

Next, we can express with a common denominator of :

This simplifies to:

Now, simplify the left side:

This combines to:

Next, we cross-multiply to eliminate the fractions:

Expanding the left side:

This simplifies to:

By setting everything to one side:

Now you can solve the quadratic equation. But before doing that, let’s check the solutions against the denominator to see if they might lead to division by zero. The denominator cannot equal zero, which implies .

Using the quadratic formula , we calculate:

The discriminant is positive (which can be computed for finding the roots), indicating there are two real solutions, and the quadratic term does not make the original equation non-valid.

However, if upon solving you reach as a potential root, it means that there is no valid solution since it would make the denominator zero.

Thus, the equation does indeed have solutions except for the value that would make the denominator zero, ensuring valid outputs, confirming that the original equation indeed has solutions.

Final answer: Yes, the equation has a solution except .