Directions: Analyze the pair of quadratic equations below to determine the relationship between their variables. Solve both equations and select the appropriate relationship.

I. $5x^2-18x+9=0$

II. $3y^2-5y-12=0$

The correct answer is x ≥ y.

To determine the relationship between the variables x and y, we solve both quadratic equations step-by-step to find their respective roots and then compare the roots of both equations.

Step 1: Solve Equation I for x

The first quadratic equation is given as:

$5x^2-18x+9=0$

We can solve this quadratic equation by splitting the middle term. We look for two numbers whose product is
$5\times 9=45$
and whose sum is
$-18$
. These two numbers are
$-15$
and
$-3$
.

Splitting the middle term:

$5x^2-15x-3x+9=0$

Factoring by grouping:

$5x(x-3)-3(x-3)=0$

$(5x-3)(x-3)=0$

Setting each factor equal to zero gives the values for x:

$5x-3=0\Rightarrow x=\frac{3}{5}=0.6$

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

Thus, the roots for x are
$x=3$
and
$x=0.6$
.

Step 2: Solve Equation II for y

The second quadratic equation is given as:

$3y^2-5y-12=0$

We split the middle term by finding two numbers whose product is
$3\times (-12)=-36$
and whose sum is
$-5$
. These numbers are
$-9$
and
$4$
.

Splitting the middle term:

$3y^2-9y+4\mathrm{y<mo>-</mo><mn>12</mn><mo>=</mo><mn>0</mn>}$

Factoring by grouping:

$3y(y-3)+4(y-3)=0$

$(3y+4)(y-3)=0$

Setting each factor equal to zero gives the values for y:

$3y+4=0\Rightarrow y=-\frac{4}{3}\approx -1.33$

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

Thus, the roots for y are
$y=3$
and
$y=-1.33$
.

Step 3: Compare the values of x and y

Comparing the values of x and y:

1. When $x=3$ and $y=3$, we have $x=y$.

2. When $x=3$ and $y=-1.33$, we have $x>y$.

3. When $x=0.6$ and $y=-1.33$, we have $x>y$.

Combining these relations (
$x=y$
and
$x>y$
), we get:

$x\ge y$