Solve the Quadratic Puzzle: 4/9x^2 + 8/3x + 4 = 0

Solve the following equation:

$\frac{4}{9}x^2+\frac{8}{3}x+4=0$

To solve the given quadratic equation $\frac{4}{9}x^2+\frac{8}{3}x+4=0$, we will apply the Quadratic Formula. Let's outline the steps.

• Step 1: Identify the coefficients.
  $a = \frac{4}{9}, \, b = \frac{8}{3}, \, c = 4$.
• Step 2: Calculate the discriminant.
  $\Delta = b^2 - 4ac = \left(\frac{8}{3}\right)^2 - 4\left(\frac{4}{9}\right)(4)$.

Calculating $\Delta$:
$\left(\frac{8}{3}\right)^2 = \frac{64}{9}$
$4 \times \frac{4}{9} \times 4 = \frac{64}{9}$,
so, $\Delta = \frac{64}{9} - \frac{64}{9} = 0$.

• Step 3: Use the Quadratic Formula. Since the discriminant is zero, there's a single solution.
• The formula simplifies to $x = \frac{-b}{2a}$.
• Calculate $x = \frac{-\frac{8}{3}}{2 \times \frac{4}{9}} = \frac{-\frac{8}{3}}{\frac{8}{9}} = -3$.

Therefore, the solution to the equation is $x = -3$.