George was assigned the function: $$ f(x) = 3x^2 − 1 $$ He was tasked with finding the real number $a$ for which the equation $$ f(4a) − f(a + 4) = 45a^2 $$ holds.
George solved the task as follows:
1) He first calculated the value of the function at $x = 4a$: $$ f(4a) = 3(4a)^2 − 1 = 3 \cdot 16a^2 − 1 = 48a^2 − 1 $$
2) Then, he calculated the value of the function at $x = a + 4$: $$ f(a + 4) = 3a^2 − 1 + 4 = 3a^2 + 3 $$
3) He subtracted the values calculated in the first two steps and got: $$ 48a^2 − 1 − 3a^2 − 3 = 45a^2 − 4 $$
4) Next, he set that difference equal to $45a^2$ and obtained the equation: $$ 45a^2 − 4 = 45a^2 $$ Finally, he stated that this equation has no solution. George claimed that the sought real number $a$ does not exist.
Did George make any mistakes? If so, identify where.
Yes, there is a mistake in step (1). The correct calculation should be: $$ f(4a) = 3 \cdot 4 \cdot (a)^2 − 1 = 12a^2 − 1 $$
Yes, there is a mistake in step (1). The correct calculation should be: $$ f(4a) = 3 \cdot 4a − 1 = 12a − 1 $$
Yes, there is a mistake in step (2). The correct calculation should be: $$ f(a + 4) = 3(a + 4)^2 − 1 = 3(a^2 + 8a + 16) − 1 = 3a^2 + 24a + 47 $$
Yes, there is a mistake in step (3). The result of subtracting $(48a^2 − 1)$ and $(3a^2 + 3)$ is incorrect.
Yes, there is a mistake in step (4). The solution to the equation $45a^2 − 4 = 45a^2$ is any real number $a$.
No, the whole procedure is correct.
Step (1) is correct: $$ f(4a) = 48a^2 − 1. $$ In step (2), the correct calculation should have been: $$ f(a + 4) = 3(a + 4)^2 − 1 = 3(a^2 + 8a + 16) − 1 = 3a^2 + 24a + 47$$ After substituting the calculated values from the first two steps into the given equation: $$ f(4a) − f(a + 4) = 45a^2 $$ we get: $$ \begin{gather} (48a^2 − 1) − (3a^2 + 24a + 47) = 45a^2 \cr 45a^2 − 24a − 48 = 45a^2 \cr −24a = 48 \cr a = −2 \end{gather} $$