Final answer:
To find the two possible values of the constant a in the expansion of (2 + x)(3 - 4x), we set the coefficient of x' equal to 30 and solve for a. The two possible values of a are -2.5 and -1/4.
Step-by-step explanation:
To find the two possible values of the constant a, we can use the given information that the coefficient of x' in the expansion of (2 + x)(3 - 4x) equals 30.
Expanding the expression (2 + x)(3 - 4x) gives us 6 + 2x - 8x - 4x^2.
Equate the coefficient of x', which is -12a, to 30 and solve for a. We get -12a = 30, which results in a = -2.5. Therefore, one possible value of a is -2.5.
To find the second possible value of a, we use the quadratic equation. Setting -4x^2 + 2x - 2.5 = 0 and solving for x gives us two values, x = -1/4 and x = 2/4. These correspond to the roots of the quadratic equation. Therefore, the second possible value of a is -1/4.