Answer:
-6(x^2 + 8x + 16) = 0
Step-by-step explanation:
Here's how to complete the square for the equation -6x^2 - 48x = -384 and identify the intermediate step:
Step 1: Move the constant term to the right side of the equation.
-6x^2 - 48x = -384
-6x^2 - 48x + 384 = 0 (Add 384 to both sides)
Step 2: Factor out a -6 from the left side.
-6(x^2 + 8x) = 0
Step 3: We want to complete the square within the parenthesis (x^2 + 8x).
* Notice that half of the coefficient of our x term (8) is 4.
* Squaring 4 gives us 16.
Incomplete square: x^2 + 8x
Complete square: (x + 4)^2
Step 4: Add and subtract the value we used to complete the square (16) on both sides of the equation to maintain equivalence.
-6(x^2 + 8x + 16 - 16) = 0
-6(x + 4)^2 + 96 = 0 (Add 96 to both sides)
Intermediate Step:
This step is reached after adding the term to complete the square but before subtracting it to maintain the equation's equivalence. So the intermediate step is:
-6(x^2 + 8x + 16) = 0
Step 5: Simplify the expression.
-6(x + 4)^2 = -96
Final Answer:
The completed square form of the equation is:
-6(x + 4)^2 = -96
This demonstrates that completing the square involves manipulating the expression to create a perfect square trinomial.
