Directions: Solve the two equations and compare the values of x and y.

I. $2x^2-14x+24=0$

II. $y^2+5y+6=0$

Correct Answer: If x is greater than y

To compare the values of x and y, we need to solve both quadratic equations step-by-step.

Step 1: Solve Equation I for x

The first equation is given as:

$2x^2-14x+24=0$

Divide the entire equation by 2 to simplify:

$x^2-7x+12=0$

Factorize the quadratic equation by finding two numbers that multiply to give 12 and add up to -7. These numbers are -3 and -4:

$x^2-3x-4x+12=0$

$x(x-3)-4(x-3)=0$

$(x-3)(x-4)=0$

Set each factor equal to zero to find the roots for x:

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

$x-4=0\Rightarrow x=4$

Thus, the values of x are 3 and 4.

Step 2: Solve Equation II for y

The second equation is given as:

$y^2+5y+6=0$

Factorize the quadratic equation by finding two numbers that multiply to give 6 and add up to 5. These numbers are 2 and 3:

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

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

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

Set each factor equal to zero to find the roots for y:

$y+2=0\Rightarrow y=-2$

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

Thus, the values of y are -2 and -3.

Step 3: Compare the values of x and y

Comparing each value of x with each value of y:
- For x = 3: 3 > -2 and 3 > -3
- For x = 4: 4 > -2 and 4 > -3

In all cases, every value of x is strictly greater than every value of y.

Hence, the correct conclusion is If x is greater than y.