Given the two quadratic equations:

I. $ x^2 - 17x - 84 = 0 $

II. $ y^2 + 4y - 117 = 0 $

Solve both equations and determine the correct relationship between $ x $ and $ y $.

Question

Given the two quadratic equations:

I. $ x^2 - 17x - 84 = 0 $

II. $ y^2 + 4y - 117 = 0 $

Solve both equations and determine the correct relationship between $ x $ and $ y $.

Accepted Answer

**Solution:**

1. Solve equation I: $ x^2 - 17x - 84 = 0 $

- Factorize by splitting the middle term:
     $$
     x^2 - 17x - 84 = x^2 + 4x - 21x - 84 = (x + 4)(x - 21) = 0
     $$
   - Roots are $ x = -4 $ and $ x = 21 $.

2. Solve equation II: $ y^2 + 4y - 117 = 0 $

- Factorize by splitting the middle term:
     $$
     y^2 + 4y - 117 = y^2 - 9y + 13y - 117 = (y - 9)(y + 13) = 0
     $$
   - Roots are $ y = 9 $ and $ y = -13 $.

3. Compare possible values of $ x $ and $ y $:

- $ x $ can be $ -4 $ or $ 21 $
   - $ y $ can be $ 9 $ or $ -13 $

4. Possible comparisons:

- $ x = 21 $ and $ y = 9 $ implies $ x > y $
   - $ x = -4 $ and $ y = 9 $ implies $ y > x $
   - $ x = 21 $ and $ y = -13 $ implies $ x > y $
   - $ x = -4 $ and $ y = -13 $ implies $ x > y $

Since $ x $ and $ y $ can take values such that sometimes $ x > y $ and sometimes $ y > x $, no definite relation can be established.

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

Answer

Answer

Answer

Answer