**Given the two quadratic equations:**

(i) $3x^2 + 11x + 10 = 0$

(ii) $2y^2 + 11y + 14 = 0$

**If $x$ and $y$ are roots of the respective equations, which of the following is true?**

Question

**Given the two quadratic equations:**

(i) $3x^2 + 11x + 10 = 0$

(ii) $2y^2 + 11y + 14 = 0$

**If $x$ and $y$ are roots of the respective equations, which of the following is true?**

Accepted Answer

**Solution:**

1. Solve the first quadratic equation:

$$3x^2 + 11x + 10 = 0$$

Factorize by splitting the middle term:

$$3x^2 + 6x + 5x + 10 = 0$$

Group terms:

$$3x(x + 2) + 5(x + 2) = 0$$

$$(3x + 5)(x + 2) = 0$$

So, the roots are:

$$x = -2, \quad x = -\frac{5}{3}$$

2. Solve the second quadratic equation:

$$2y^2 + 11y + 14 = 0$$

Factorize by splitting the middle term:

$$2y^2 + 7y + 4y + 14 = 0$$

Group terms:

$$y(2y + 7) + 2(2y + 7) = 0$$

$$(y + 2)(2y + 7) = 0$$

So, the roots are:

$$y = -2, \quad y = -\frac{7}{2}$$

3. Now, compare the roots:

- For $x$: $-2, -\frac{5}{3} \approx -1.666$
   - For $y$: $-2, -\frac{7}{2} = -3.5$

4. The largest root of $x$ is $-1.666$ and the largest root of $y$ is $-2$.

5. Since $-1.666 > -2$, the maximum $x$ root is greater than or equal to the maximum $y$ root.

**Answer:** (a) $x ≥ y$

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