Question

The owner of a video game store creates the expression −2x2 + 32x − 8 to represent the store's weekly profit in dollars, where x represents the price of a new video game. Choose the equivalent expression that reveals the video game price that produces the highest weekly profit, and use it to determine that price.

−2(x2 − 16x) − 8; x = $16
−2(x2 − 16x + 4); x = $4
−2(x − 8)2 + 120; x = $8
−2(x − 8)2 + 120; x = $120

Answer 1

To find the equivalent expression that reveals the video game price that produces the highest weekly profit, we need to complete the square for the given expression −2x2 + 32x − 8.

Starting with the expression -2x^2 + 32x - 8, we can factor out the -2 from the x^2 and x terms:
-2(x^2 - 16x) - 8

Now, to complete the square, we need to add and subtract (16/2)^2 = 64 to the expression inside the parentheses:
-2(x^2 - 16x + 64 - 64) - 8
-2((x - 8)^2 - 64) - 8
-2(x - 8)^2 + 128 - 8
-2(x - 8)^2 + 120

Therefore, the equivalent expression is −2(x − 8)2 + 120, and the price of the video game that produces the highest weekly profit is $8.