Final answer:
To express quadratic expressions in the form of ((x + a)² + b), we complete the square for each given expression. For (x² + 12x), the result is A. ((x + 6)² - 36), equivalent to option A. For (x² + 1/2x), the result is D. ((x + 1/4)² - 1/16), equivalent to option D.
Step-by-step explanation:
The question deals with expressing quadratic expressions in the form of ((x + a)² + b). To do this, we complete the square for each given quadratic expression.
For expression a) (x² + 12x):
Take half the coefficient of x, which is 6, and square it to get 36.
Add and subtract this square within the expression to get x² + 12x + 36 - 36.
Rewrite this as ((x + 6)² - 36), which can be expressed as ((x + 6)² + (-36)).
For expression b) (x² + 1/2x):
Take half the coefficient of x, which is 1/4, and square it to get 1/16.
Add and subtract this square within the expression to get x² + 1/2x + 1/16 - 1/16.
Rewrite this as ((x + 1/4)² - 1/16), which can be expressed as ((x + 1/4)² + (-1/16)).
Therefore, the correct expressions are:
A. ((x + 6)² - 36), which is equivalent to option A.
D. ((x + 1/4)² - 1/16), which is equivalent to option D.