The discriminant of the quadratic equation is (-12)² - 4(3)(10) i.e., 24. As this is positive but not a perfect square, the roots are irrational and unequal.

Sum of the roots and the product of the roots are -20 and 3 respectively.

x² + 5x - 3x - 15 = 0
x(x + 5) - 3(x + 5) = 0
(x - 3)(x + 5) = 0
=> x = 3 or x = -5.

Let the roots of the equation 2a and 3a respectively.
2a + 3a = 5a = -(- 5/2) = 5/2 => a = 1/2
Product of the roots: 6a² = b/2 => b = 12a²
a = 1/2, b = 3.

a² + b² + ab = a² + b² + 2ab - ab
i.e., (a + b)² - ab
from x² - 9x + 20 = 0, we have
a + b = 9 and ab = 20. Hence the value of required expression (9)² - 20 = 61.

Let the roots of the quadratic equation be x and 3x.
Sum of roots = -(-12) = 12
a + 3a = 4a = 12 => a = 3
Product of the roots = 3a² = 3(3)² = 27.

Three consecutive even natural numbers be 2x - 2, 2x and 2x + 2.
(2x - 2)² + (2x)² + (2x + 2)² = 1460
4x² - 8x + 4 + 4x² + 8x + 4 = 1460
12x² = 1452 => x² = 121 => x = ± 11
As the numbers are positive, 2x > 0. Hence x > 0. Hence x = 11.
Required numbers are 20, 22, 24.

Let the two consecutive positive integers be x and x + 1
x² + (x + 1)² - x(x + 1) = 91
x² + x - 90 = 0
(x + 10)(x - 9) = 0 => x = -10 or 9.
As x is positive x = 9
Hence the two consecutive positive integers are 9 and 10.

a/b + b/a = (a² + b²)/ab = (a² + b² + a + b)/ab
= [(a + b)² - 2ab]/ab
a + b = -8/1 = -8
ab = 4/1 = 4
Hence a/b + b/a = [(-8)² - 2(4)]/4 = 56/4 = 14.

I.(a - 3)(a - 4) = 0
=> a = 3, 4
II. (b - 2)(b - 1) = 0
=> b = 1, 2
=> a > b