3x ² + 8x + 4 = 0; 4y ² - 19y + 12 = 0

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]

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 - 32 = 112
? x² = 112 + 32 ? x² = 144
? x = ± 12
and y - ?169 = 0
? y = ?169
? y = ± 13
? Relation cannot be established.

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

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

Let Mr . Arjun is on tour for N days.
Then, according to the question,
Difference in expenses per day for original and extended tour = 3
360/N - 360/(N + 4) = 3
? 1/N - 1/(N + 4) = 3/360 = 1/120
? [N + 4 - N] / [ N x (N + 4)] = 1/120
? N(N + 4) = 4 x 120 = 480
? N² + 4N - 480 = 0
? N² + 24 - 20x - 480 = 0
? N(N + 24) - 20(N + 24) = 0
? (N + 24) (N - 20) = 0
? N = 20
(we ignore -ve value of x, that is, -24 because days cannot be negative)

As per the given above question , we have
Let k₁ and k₂ be the roots of quadratic equation .
One root is k₁ = ?5
So the other root is k₂ = ? ?5
? Sum of the roots = k₁ + k₂ = ?5 + ( -?5 ) = 0
and product of the roots = k₁ × k₂ = ? (?5) (- ?5) = ?5
? Required equation is x² - (sum of the roots)x + (product of the roots) = 0
? x² - 5 = 0.