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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.