x ² - 32 = 112; y - ? 169 = 0

x² - 16 = 0
? x² = 16 ? x = ?16 = ± 4
and y² - 9y + 20 = 0
y² - 4y - 5y + 20 = 0
? y(y - 4) - 5(y - 4) = 0
? y = 5, 4
? y ? x or x ? y

x - ?121 = 0
? x = ?121 ? x = ± 11
and y² - 121 = 0 ? y² = 121
? y = ?121 = ± 11
? x = y

225x² - 4 = 0;
? 225x² = 4 ? x² = 4/225
? x = ?4/225 = ± 2/15, i.e., 2/15 and -2/15
and ?225y + 2 = 0 or ?225y = -2
On squaring both sides, we get
(?225y)² = (-2)²
? 225y = 4
? y = 4/225
So, relation cannot be established because 4/225 lies between 2/15 and -2/15.

x² + x - 20 = 0
      [by factorisation method]
? x² + 5x - 4x - 20 = 0
? x(x + 5) - 4(x + 5) = 0
(x + 5) (x - 4) = 0
? x = -5 or 4
and y² - y - 30 = 0
? y² - 6y + 5y - 30 = 0
? y(y - 6) + 5(y - 6) = 0
? (y - 6) (y + 5) = 0
? y = 6 or - 5
Hence, y ? x or x ? y

? + ? = - p, ? ? = 8
Given, ? - ? = 2
On squaring both sides, we get
(? - ?)² = 4
? (? + ?)² - 4?? = 4
? p² - 32 = 4
? p² = 36 ? p = ± 6

x² - 8x + 15 = 0
? x² - 5x - 3x + 15 = 0
? x(x - 5) -3(x - 5) = 0
? (x - 5) (x - 3) = 0
? x = 5, 3
and y² - 3y + 2 = 0
? y² - 2y - y + 2 = 0
? y(y - 2) -1(y - 2) = 0
? (y -2) (y - 1) = 0
? y = 2, 1
? x > y

x² - x - 12 = 0
? x² - 4x + 3x - 12 = 0
? x(x - 4) + 3 (x - 4) = 0
? (x - 4) (x + 3) = 0
? x = - 3, 4
and y² + 5y + 6 = 0
? y² + 3y + 2y + 6 = 0
? y(y + 3) + 2(y + 3) = 0
? (y + 3) (y + 2) = 0
? y = - 3, - 2
? x ? y
[? x = - 3 and y = - 3, so x = y and x = 4 and y = - 2, hence x > y]

3x² + 8x + 4 = 0
? 3x² + 6x + 2x + 4 = 0
? 3x(x + 2) + 2(x + 2) = 0
? (x + 2) (3x + 2) = 0
? x = - 2, - 2/3
and 4y² - 19y + 12 = 0
? 4y² - 16y - 3y + 12 = 0
? 4y(y - 4) -3(y - 4) = 0
? (y - 4) (4y - 3) = 0
? y = 4, 3/4
Hence, y > x or x < y

x² - 365 = 364
? x² = 364 + 365
? x = ?729 = ± 27
and y - ?324 = ?81 ? y - 18 = 9
? y = 27
So, y ? x or x ? y because y = 27 and x = 27 and - 27.

4/?x + 7/?x = ?x;
? 11/?x = ?x
? x = 11
and y² - (11)^5/2/?y = 0
? y² = (11)^5/2/(y)^1/2
? y² x y^1/2 = (11)^5/2
? (y)^5/2 = (11)^5/2
? y = 11
? x = y