Given the two quadratic equations:

(i) $6x^2 - 11x + 4 = 0$

(ii) $3y^2 - 5y + 2 = 0$

Determine the correct relationship between the roots $x$ and $y$:

Question

Given the two quadratic equations:

(i) $6x^2 - 11x + 4 = 0$

(ii) $3y^2 - 5y + 2 = 0$

Determine the correct relationship between the roots $x$ and $y$:

Accepted Answer

**Solution:**

1. Solve the first equation:

$6x^2 - 11x + 4 = 0$

Factorizing:

$$6x^2 - 8x - 3x + 4 = 0$$

$$(2x)(3x - 4) - 1(3x - 4) = 0$$

$$(2x - 1)(3x - 4) = 0$$

So, $x = \frac{1}{2}$ or $x = \frac{4}{3}$.

2. Solve the second equation:

$3y^2 - 5y + 2 = 0$

Factorizing:

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

$$3y(y - 1) - 2(y - 1) = 0$$

$$(3y - 2)(y - 1) = 0$$

So, $y = \frac{2}{3}$ or $y = 1$.

3. Compare the roots:

- $x$ values: $0.5, 1.333...$

- $y$ values: $0.666..., 1$

Since some $x$ roots are less than some $y$ roots and some are greater, no consistent relation like $x < y$ or $x > y$ can be established.

**Answer:** (e) $x = y$ or no relation can be established between $x$ and $y$.

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