Given the two quadratic equations:

I. $(x^2 + 12x + 35 = 0)$

II. $(y^2 + 7y + 10 = 0)$

Which of the following relations between $x$ and $y$ is correct?

Question

Given the two quadratic equations:

I. $(x^2 + 12x + 35 = 0)$

II. $(y^2 + 7y + 10 = 0)$

Which of the following relations between $x$ and $y$ is correct?

Accepted Answer

**Solution:**

1. Consider the first quadratic equation:

$$ x^2 + 12x + 35 = 0 $$

Factorize it by splitting the middle term:

$$ x^2 + 5x + 7x + 35 = 0 $$

Group terms:

$$ x(x + 5) + 7(x + 5) = 0 $$

Factor out common binomial:

$$ (x + 5)(x + 7) = 0 $$

So, the roots are:

$$ x = -5, \quad x = -7 $$

2. Now, consider the second quadratic equation:

$$ y^2 + 7y + 10 = 0 $$

Factorize by splitting the middle term:

$$ y^2 + 2y + 5y + 10 = 0 $$

Group terms:

$$ y(y + 2) + 5(y + 2) = 0 $$

Factor out common binomial:

$$ (y + 2)(y + 5) = 0 $$

So, the roots are:

$$ y = -2, \quad y = -5 $$

3. Now, compare the roots:

- The roots of $x$ are $-7$ and $-5$.
   - The roots of $y$ are $-5$ and $-2$.

The maximum root of $x$ is $-5$, and the minimum root of $y$ is $-5$.

Since $-7 \leq -5 \leq -2$, the values of $x$ are less than or equal to the values of $y$.

**Answer:** $x \leq y$

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