Given the two quadratic equations:

I. $( x^2 - 15x + 56 = 0 )$

II. $( y^2 - 8y + 15 = 0 )$

Which of the following relations between $ x $ and $ y $ is true?

Question

Given the two quadratic equations:

I. $( x^2 - 15x + 56 = 0 )$

II. $( y^2 - 8y + 15 = 0 )$

Which of the following relations between $ x $ and $ y $ is true?

Accepted Answer

**Solution:**

1. Consider the first equation:
   $$
   x^2 - 15x + 56 = 0
   $$
   Factorizing:
   $$
   x^2 - 7x - 8x + 56 = 0 \
   x(x - 7) - 8(x - 7) = 0 \
   (x - 7)(x - 8) = 0
   $$
   So, the roots are $ x = 7 $ and $ x = 8 $.

2. Now, consider the second equation:
   $$
   y^2 - 8y + 15 = 0
   $$
   Factorizing:
   $$
   y^2 - 5y - 3y + 15 = 0 \
   y(y - 5) - 3(y - 5) = 0 \
   (y - 5)(y - 3) = 0
   $$
   So, the roots are $ y = 5 $ and $ y = 3 $.

3. Comparing the roots:
   - The smallest root of $ x $ is 7, which is greater than the largest root of $ y $, which is 5.

Therefore, all values of $ x $ are greater than all values of $ y $.

**Answer:** $ x > y $

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