Introduction:

We are given the quadratic expression ax^2 |2a - 3|x - 6| and we need to find the number of integral values of 'a' for which this expression is positive for exactly two integral values of x.

Analysis:

To solve this problem, we need to analyze the behavior of the given quadratic expression for different values of 'a' and 'x'. Let's break down the analysis into two cases:

Case 1: a ≥ 0
In this case, the expression ax^2 |2a - 3|x - 6| will always be non-negative for all values of x because the quadratic term is always non-negative for a ≥ 0. Therefore, there are no integral values of 'a' that satisfy the condition of exactly two positive values.

Case 2: a < />
In this case, the expression ax^2 |2a - 3|x - 6| can be positive for some values of x. To find the values of 'a' for which the expression is positive for exactly two integral values of x, we need to consider the following scenarios:

Scenario 1: 2a - 3 ≥ 0
In this scenario, the quadratic term ax^2 is non-negative for all values of x, and the expression can only be positive when |x - 6| = 1. This implies that x = 5 or x = 7. Therefore, in this case, the expression is positive for exactly two integral values of x.

Scenario 2: 2a - 3 < />
In this scenario, the quadratic term ax^2 is negative for some values of x. To make the expression positive for exactly two integral values of x, we need to consider two sub-scenarios:

Sub-scenario 2.1: x - 6 ≥ 0
In this sub-scenario, the quadratic term ax^2 is positive for all x ≥ 6. To make the expression positive for exactly two integral values of x, we need to find the values of 'a' for which the quadratic term changes sign twice. This happens when the discriminant of ax^2 is positive, i.e., 4a > 0. Therefore, in this sub-scenario, the expression is positive for all a > 0.

Sub-scenario 2.2: x - 6 < />
In this sub-scenario, the quadratic term ax^2 is negative for all x < 6.="" to="" make="" the="" expression="" positive="" for="" exactly="" two="" integral="" values="" of="" x,="" we="" need="" to="" find="" the="" values="" of="" 'a'="" for="" which="" the="" quadratic="" term="" changes="" sign="" twice.="" this="" happens="" when="" the="" discriminant="" of="" ax^2="" is="" positive,="" i.e.,="" 4a="" /> 0. Therefore, in this sub-scenario, the expression is positive for all a > 0.

Final Answer:

From the analysis above, we can conclude that there are no integral values of 'a' for which the quadratic expression ax^2 |2a - 3|x - 6| is positive for exactly two integral values of x.