Directions: Each question below contains two quantities, labeled Quantity I and Quantity II. Determine the relationship between the two quantities and select the appropriate option. Compare numerical values only.

If $x^2-6x+8=0$ and $2y^2-7y+6=0$, then
Quantity I: Product of the largest root of both equations
Quantity II: $\sqrt[3]{512}$

The correct option is Quantity I = Quantity II or no relation.

Let us evaluate Quantity I and Quantity II step-by-step.

Step 1: Find the roots of the first quadratic equation.

The given equation for x is:

$x^2-6x+8=0$

Factoring the quadratic expression:

$x^2-4x-2x+8=0$

$x(x-4)-2(x-4)=0$

$(x-2)(x-4)=0$

Solving for x gives:

$x=2$ or $x=4$

The largest root of the first equation is $x=4$.

Step 2: Find the roots of the second quadratic equation.

The given equation for y is:

$2y^2-7y+6=0$

Factoring the quadratic expression:

$2y^2-4y-3y+6=0$