Final answer:
By using the formula for the difference between the roots of a quadratic equation, we find that the coefficient c must satisfy the equation 9c^2 + 48c = 0 for the differences between the roots of the two given equations to be equal.
Step-by-step explanation:
The question asks us to find the value for which the difference between the roots of two quadratic equations is equal. The first quadratic equation is 3x^2 - 2x + c = 0 and the second is 2x^2 - cx + 3 = 0.
To solve for this, we need to use the formula for the roots of a quadratic equation, which is x = (-b ± √(b^2 - 4ac)) / (2a). The difference between the roots of a quadratic equation ax^2 + bx + c = 0 is given by the formula √(b^2 - 4ac) / |a|, which comes from subtracting the two root expressions derived from the quadratic formula.
For the first equation, the difference between the roots is √(2^2 - 4*3*c) / 3, and for the second equation, it is √((c^2) - 4*2*3) / 2. Setting these two expressions equal to each other and solving for c, we eventually obtain 9c^2 + 48c = 0 as the requested expression, which indicates that the coefficient c must satisfy this equation for the root differences to be equal.