Consider the quadratic equation ax2 − bx + c  =  0, a, b, c  ∈  N  , , which has two distinct real roots belonging to the interval (1,2)  .The least value of a is

The problem is asking for the least value of 'a' in the quadratic equation ax^2 - bx + c = 0, given that the equation has two distinct real roots that fall within the interval (1,2).

Step 1: Understand the problem The roots of the quadratic equation are the x-values where the parabola crosses the x-axis. In this case, we know that these roots are between 1 and 2.

Step 2: Apply the quadratic formula The quadratic formula is used to find the roots of a quadratic equation. It is given by:

x = [ -b ± sqrt(b^2 - 4ac) ] / 2a

Step 3: Apply the conditions of the problem We know that the roots are distinct and real, which means the discriminant (b^2 - 4ac) must be greater than 0.

Step 4: Find the least value of 'a' To minimize 'a', we need to maximize the denominator of the quadratic formula. This happens when the roots are at the extremes of the interval (1,2), i.e., when the roots are 1 and 2.

So, we have two equations from the roots of the quadratic equation:

a + b + c = 0 (from root 1) 4a + 2b + c = 0 (from root 2)

Solving these equations, we get a = 1, b = -3, and c = 2.

Therefore, the least value of 'a' is 1.