Given the two quadratic equations:

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

(ii) $ y^2 - 5y + 4 = 0 $

Determine the correct relationship between $x$ and $y$ from the following options:

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Question

Given the two quadratic equations:

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

(ii) $ y^2 - 5y + 4 = 0 $

Determine the correct relationship between $x$ and $y$ from the following options:

`

Accepted Answer

**Solution:**

1. Solve the first equation:

$$
2x^2 + 2x - 4 = 0
$$
Divide the entire equation by 2:

$$
x^2 + x - 2 = 0
$$
Factorizing:

$$
(x + 2)(x - 1) = 0
$$
So, the roots are:

$$
x = -2, \quad x = 1
$$

2. Solve the second equation:

$$
y^2 - 5y + 4 = 0
$$
Factorizing:

$$
(y - 4)(y - 1) = 0
$$
So, the roots are:

$$
y = 4, \quad y = 1
$$

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

- $x$ can be $-2$ or $1$
- $y$ can be $1$ or $4$

4. Check all possible comparisons:

- $-2 < 1$, $1 = 1$
- $1 < 4$, $-2 < 4$

In all cases, $x \leq y$.

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

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