Answer:
The total profits will be zero at sales prices of $4 and $9
Explanation:
∵ The function is y = -3x² + 39x - 108, where
- y is the total profits of hundreds of dollars
The zeroes of the function are the values of x at y = 0
→ To find the zeroes of the function above equate y by 0
∵ 0 = -3x² + 39x - 108
→ Switch the two sides
∴ -3x² + 39x - 108 = 0
→ Divide all terms by -3 to simplify the equation
∴ x² - 13x + 36 = 0
→ Let us factor the trinominal into two brackets
∵ The last term is +ve and the middle term is -ve
∴ The two brackets have (-) sign in the middle ⇒ (... - ...)(... - ...)
∵ x² = (x)(x)
∵ 36 = (4)(9)
∵ (4)(x) + (9)(x) = 4x + 9x = 13x ⇒ the value of the middel term
→ That means the factors of the trinomial are (x - 4) and (x - 9)
∴ x² - 13x + 36 = (x - 4)(x - 9)
→ Replace the trinomial by its factors
∴ (x - 4)(x - 9) = 0
→ Equate each bracket by 0 to find x
∵ x - 4 = 0
→ Add 4 to both sides
∴ x = 4
∵ x - 9 = 0
→ Add 4 to both sides
∴ x = 9
∴ The zeroes of the functions are 4 and 9
∵ x represents the sales price in dollars
∴ The sales prices which make zero profit are $4 and $9
∴ The total profits will be zero at sales prices of $4 and $9