Solve the Quadratic Equation with Fractional Terms: x² + 10/9x + 25/81 = 0

To solve the given quadratic equation $x^2 + \frac{10}{9}x + \frac{25}{81} = 0$, we will first check if it can be expressed as a perfect square trinomial.

Notice that a quadratic equation in the form $(x + d)^2 = 0$ expands to:

$(x + d)^2 = x^2 + 2dx + d^2$.

We observe:

• $2d = \frac{10}{9}$, which gives $d = \frac{10}{18} = \frac{5}{9}$.
• $d^2 = \left(\frac{5}{9}\right)^2 = \frac{25}{81}$.

This confirms that the original equation can be rewritten as:

$(x + \frac{5}{9})^2 = 0$.

Solving $(x + \frac{5}{9})^2 = 0$ yields:

$x + \frac{5}{9} = 0$.

Subtracting $\frac{5}{9}$ from both sides, we get:

$x = -\frac{5}{9}$.

Therefore, the solution to the equation is $x = -\frac{5}{9}$, which corresponds to choice id="3".