In each of these questions, two equations (I) and (II) are given. You have to solve both the equations and give answers.

$\text{I.}{x}^2+12x+35=0$

$\text{II.}{y}^2+7y+10=0$

The correct answer is $x\le y$.

To establish the relationship between $x$ and $y$, we solve both quadratic equations step-by-step.

Step 1: Solve Equation I for x

The first equation is:
$x^2+12x+35=0$

We look for two numbers that multiply to 35 and add up to 12. These numbers are 7 and 5.
Splitting the middle term:
$x^2+7x+5x+35=0$
$x(x+7)+5(x+7)=0$
$(x+7)(x+5)=0$
Thus, the roots for $x$ are:
$x=-7$
and
$x=-5$

Step 2: Solve Equation II for y

The second equation is:
$y^2+7y+10=0$

We look for two numbers that multiply to 10 and add up to 7. These numbers are 5 and 2.
Splitting the middle term:
$y^2+5y+2y+10=0$
$y(y+5)+2(y+5)=0$
$(y+5)(y+2)=0$
Thus, the roots for $y$ are:
$y=-5$
and
$y=-2$

Step 3: Compare the values of x and y

Now, let us compare every root of $x$ with every root of $y$:
• For $x=-5$ and $y=-5$: $x=y$
• For $x=-5$ and $y=-2$: $x<y$
• For $x=-7$ and $y=-5$: $x<y$
• For $x=-7$ and $y=-2$: $x<y$

Combining all case comparisons, we conclude that:
$x\le y$