Problem 154 - College Algebra Essentials

Question

Perform the indicated operation or operations. $$(f(x))^{2}-2 f(x)+6, \text { where } f(x)=3 x-4$$

Step-by-Step Solution

Answer

The result of the operation is \(9x^2 - 30x + 30\).

Step 1: Substitution

First, substitute the function \(f(x) = 3x - 4\) into the expression. This results in \((3x - 4)^2 - 2(3x - 4) + 6\).

Step 2: Expanding the Square Term

Expand the square term. Remember that \((a - b)^2 = a^2 - 2ab + b^2\). This gives us \((3x)^2 - 2*(3x)*(4) + (4)^2 - 2(3x - 4) + 6\). After simplification, this gives \(9x^2 - 24x + 16 - 2(3x - 4) + 6\).

Step 3: Distribute the -2

Next, distribute the -2 in the term \( - 2(3x - 4)\). This yields \(9x^2 - 24x + 16 - 6x + 8 + 6\).

Step 4: Simplify the Expression

Finally, combine like terms to get the final expression: \(9x^2 - 30x + 30\).