Given the following equations: (i) $ x^2 + 6x - 16 = 0 $ (ii) $ y^2 - 6y + 5 = 0 $ Determine the correct relationship between $ x $ and $ y $. Choose: - (a) if $ x y $ - (b) if $ x \geq y $ - (c) if $ x

Question

Given the following equations:

(i) $ x^2 + 6x - 16 = 0 $

(ii) $ y^2 - 6y + 5 = 0 $

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

Choose:
- (a) if $ x > y $
- (b) if $ x \geq y $
- (c) if $ x < y $
- (d) if $ x \leq y $
- (e) if $ x = y $ or the relationship cannot be established

Accepted Answer

**Solution:**

1. Solve the first quadratic equation:
   $$
   x^2 + 6x - 16 = 0
   $$
   Factorizing:
   $$
   x^2 + 8x - 2x - 16 = 0 \ 
   x(x + 8) - 2(x + 8) = 0 \ 
   (x - 2)(x + 8) = 0
   $$
   So, $ x = 2 $ or $ x = -8 $.

2. Solve the second quadratic equation:
   $$
   y^2 - 6y + 5 = 0
   $$
   Factorizing:
   $$
   y^2 - 5y - y + 5 = 0 \ 
   y(y - 5) - 1(y - 5) = 0 \ 
   (y - 1)(y - 5) = 0
   $$
   So, $ y = 1 $ or $ y = 5 $.

3. Possible values of $ x $ are $ 2 $ and $ -8 $, and possible values of $ y $ are $ 1 $ and $ 5 $.

4. Comparing these values:
   - $ x = 2 $ can be greater than $ y = 1 $ but less than $ y = 5 $.
   - $ x = -8 $ is less than both $ y = 1 $ and $ y = 5 $.

5. Since $ x $ and $ y $ can take multiple values and no definite relationship holds for all values, the relationship between $ x $ and $ y $ cannot be established.

**Answer:** (e) $ x = y $ or the relationship cannot be established

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