Solve the Fraction Equation: x^2/3 + 4/3 = 3(x - 2)(x + 2)

To solve the equation $\frac{x^2 + 4}{3} = 3(x-2)(x+2)$, we will follow these steps:

• Step 1: Remove the fraction by multiplying both sides by 3.
• Step 2: Expand and simplify the right side using the difference of squares.
• Step 3: Rearrange into a standard quadratic form.
• Step 4: Solve the quadratic equation.

Let's begin:

Step 1: Multiply both sides by 3 to eliminate the fraction:

$x^2 + 4 = 9(x-2)(x+2)$

Step 2: Recognize that $(x-2)(x+2)$ is a difference of squares:

$(x-2)(x+2) = x^2 - 4$

Then we have:

$x^2 + 4 = 9(x^2 - 4)$

Step 3: Distribute the 9 on the right side:

$x^2 + 4 = 9x^2 - 36$

Step 4: Rearrange this into a standard quadratic form:

$x^2 - 9x^2 = -36 - 4$

Simplify:

$-8x^2 = -40$

Divide everything by -8 to solve for $x^2$:

$x^2 = 5$

Step 5: Solve by taking the square root of both sides:

$x = \pm \sqrt{5}$

Therefore, the solution to the problem is $x = \pm \sqrt{5}$.