Solve the following quadratic equations and determine the correct relationship between *x* and *y*:

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

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

Which of the following is true?

Question

Solve the following quadratic equations and determine the correct relationship between *x* and *y*:

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

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

Which of the following is true?

Accepted Answer

**Solution:**

1. Solve equation (I): $$x^2 + 12x + 35 = 0$$

- Calculate the discriminant:
     $$D = 12^2 - 4 	imes 1 	imes 35 = 144 - 140 = 4$$

- Roots are:
     $$x = \frac{-12 \pm \sqrt{4}}{2} = \frac{-12 \pm 2}{2}$$

- So,
     $$x_1 = \frac{-12 + 2}{2} = \frac{-10}{2} = -5$$
     $$x_2 = \frac{-12 - 2}{2} = \frac{-14}{2} = -7$$

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

- Calculate the discriminant:
     $$D = 7^2 - 4 	imes 1 	imes 10 = 49 - 40 = 9$$

- Roots are:
     $$y = \frac{-7 \pm \sqrt{9}}{2} = \frac{-7 \pm 3}{2}$$

- So,
     $$y_1 = \frac{-7 + 3}{2} = \frac{-4}{2} = -2$$
     $$y_2 = \frac{-7 - 3}{2} = \frac{-10}{2} = -5$$

3. Now, compare the roots of *x* and *y*:

- *x* roots: -5, -7
   - *y* roots: -2, -5

4. To find the relationship between *x* and *y*, consider all pairs:

- Compare largest roots: -5 (x) and -2 (y) → -5 < -2
   - Compare smallest roots: -7 (x) and -5 (y) → -7 < -5

5. Since both roots of *x* are less than or equal to the roots of *y*, the correct relation is:

$$x \leq y$$

**Answer:** (b) $$x \leq y$$

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