The area of a rectangle can be represented by: \( 3 x^{2}-4 x-32 \) (ii) Given that this quadratic expression represents the area of a rectangle, what would be the possible values of \( x \) ? Justify your answer:

To find the possible values of \( x \) that make the area of the rectangle valid, we need to determine the roots of the quadratic expression \( 3x^2 - 4x - 32 = 0 \) using the quadratic formula: \[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \] Here, \( a = 3 \), \( b = -4 \), and \( c = -32 \). Plugging in these values gives: \[ x = \frac{-(-4) \pm \sqrt{(-4)^2 - 4 \cdot 3 \cdot (-32)}}{2 \cdot 3} = \frac{4 \pm \sqrt{16 + 384}}{6} = \frac{4 \pm \sqrt{400}}{6} = \frac{4 \pm 20}{6} \] Calculating the two potential solutions, we have \( x = \frac{24}{6} = 4 \) and \( x = \frac{-16}{6} = -\frac{8}{3} \). Since \( x \) represents a physical length, it must be non-negative, thus the only possible value for \( x \) is \( 4 \) units. As for the broader implications, the quadratic function represents a physical space; thus, practical applications like architectural design or landscaping would require dimensions to always be positive. It's a fun challenge to ensure the math translates into real-world scenarios!