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

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

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

The correct option is If x is greater than or equal to y.

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

Step 1: Solve Equation I for x

The first quadratic equation is given by:

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

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

Factoring the equation:

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

Equating each factor to zero:

$x-6=0\Rightarrow x=6$

$x-3=0\Rightarrow x=3$

Thus, the values of x are 3 and 6.

Step 2: Solve Equation II for y

The second quadratic equation is given by:

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

We split the middle term by finding two numbers that multiply to 2 × 9 = 18 and add up to -9. These numbers are -6 and -3.

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

Grouping the terms:

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

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

Equating each factor to zero:

$2y-3=0\Rightarrow y=\frac{3}{2}=1.5$

$y-3=0\Rightarrow y=3$

Thus, the values of y are 1.5 and 3.

Step 3: Compare the values of x and y

Now, let's compare every value of x with every value of y:

• For x = 6: 6 > 1.5 and 6 > 3, so x > y.

• For x = 3: 3 > 1.5 and 3 = 3, so x ≥ y.

Combining all cases, we find that x is always greater than or equal to y (x ≥ y).

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