Evelyn is a graphic designer who runs an online T-shirt store. She figures out that her daily profit, in dollars, can be modeled by the function P(x)=–2x2+54x–280, where x is the price of each T-shirt, also in dollars. Evelyn will break even when her profit is $0.Rewrite the function in the form P(x)=a(x–r1)(x–r2) by factoring. What are the values of a, r1, and r2?
Question
Evelyn is a graphic designer who runs an online T-shirt store. She figures out that her daily profit, in dollars, can be modeled by the function P(x)=–2x2+54x–280, where x is the price of each T-shirt, also in dollars. Evelyn will break even when her profit is $0.Rewrite the function in the form P(x)=a(x–r1)(x–r2) by factoring. What are the values of a, r1, and r2?
Solution
The first step is to factor the quadratic equation. The equation given is P(x) = -2x^2 + 54x - 280.
We can factor out a -2 from each term to get P(x) = -2(x^2 - 27x + 140).
Next, we factor the quadratic x^2 - 27x + 140. We are looking for two numbers that multiply to 140 and add to -27. Those numbers are -20 and -7.
So, we can write the equation as P(x) = -2(x - 20)(x - 7).
Therefore, the values are a = -2, r1 = 20, and r2 = 7.
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