Final answer:
There appears to be an error with the provided question or options, as the correct value calculated for 'b' is -18, which doesn't match any of the answer choices given.
Step-by-step explanation:
If the roots of the equation 3x² + ax - 2 = 0 are equal, the discriminant must be zero. The discriminant of a quadratic equation ax² + bx + c = 0 is given by b² - 4ac. So, for the given equation, it means a² - 4(3)(-2) = 0. Solving this gives a² = 24, so a = ±6√6 (only positive value is considered as usually, the leading coefficient 'a' is considered positive).
For the quadratic equation 9(x² + 6x) - b = 0, expanding and simplifying we get 9x² + 54x - b = 0. Comparing this with the standard quadratic equation form ax² + bx + c = 0, we can see that 'a' equals 9, 'b' (the coefficient of x) equals 54, and 'c' is the term 'b' we want to find. Since 'a' and 'b' are matching coefficients in both equations, and the roots are equal, therefore, 'b' in the second equation should equal -2 * 9 (from the original equation). Hence, b = -18. However, none of the provided options (A) b = 54 (B) b = 48 (C) b = 36 (D) b = 24 have this value, suggesting a possible error in the question or options.