Solve the following two quadratic equations to find the correct relationship between the variables x and y.

$\text{I.}{x}^2-9x+18=0$

$\text{II.}2y^2-9y+9=0$

Correct Option: If x is greater than or equal to y

To find the relationship between x and y, we need to solve both quadratic equations step-by-step.

Step 1: Solve Equation I for x

$x^2-9x+18=0$

We need to find two numbers that multiply to give +18 and add up to -9. These numbers are -3 and -6.

$x^2-3x-6x+18=0$

$x(x-3)-6(x-3)=0$

$(x-3)(x-6)=0$

Therefore, the values of x are:

$x=3$ or $x=6$

Step 2: Solve Equation II for y

$2y^2-9y+9=0$

We need two numbers that multiply to give $2\times 9=18$ and add up to -9. These numbers are -3 and -6.

$2y^2-3y-6y+9=0$

$y(2y-3)-3(2y-3)=0$

$(2y-3)(y-3)=0$

Therefore, the values of y are:

$y=\frac{3}{2}=1.5$ or $y=3$

Step 3: Compare the values of x and y

Let's compare each value of x with all values of y:

- For $x=3$: when $y=1.5$, $x>y$; when $y=3$, $x=y$.
- For $x=6$: when $y=1.5$, $x>y$; when $y=3$, $x>y$.

In all cases, $x$ is either greater than or equal to $y$ ($x\ge y$).

Hence, the correct relationship is If x is greater than or equal to y.