Given the equations:

(i) $ x^2 - 11x + 30 = 0 $

(ii) $ y^2 + y - 20 = 0 $

Determine the correct relationship between $ x $ and $ y $.

Question

Given the equations:

(i) $ x^2 - 11x + 30 = 0 $

(ii) $ y^2 + y - 20 = 0 $

Determine the correct relationship between $ x $ and $ y $.

Accepted Answer

**Solution:**

1. Solve the first quadratic equation for $ x $:

$$
   x^2 - 11x + 30 = 0
   $$

Factorizing:

$$
   x^2 - 6x - 5x + 30 = 0 \
   x(x - 6) - 5(x - 6) = 0 \
   (x - 6)(x - 5) = 0
   $$

So, $ x = 6 $ or $ x = 5 $.

2. Solve the second quadratic equation for $ y $:

$$
   y^2 + y - 20 = 0
   $$

Factorizing:

$$
   y^2 + 5y - 4y - 20 = 0 \
   y(y + 5) - 4(y + 5) = 0 \
   (y - 4)(y + 5) = 0
   $$

So, $ y = 4 $ or $ y = -5 $.

3. Compare the possible values:

- Minimum $ x $ is 5, maximum $ y $ is 4.

- Since even the smallest $ x $ (5) is greater than the largest $ y $ (4), it follows that $ x > y $.

**Answer:** (a) $ x > y $

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