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Cut Safe company manufactures scissors. The company uses the function P (s)=-25s^(2)+250 s-400 to model its profit, where P(x) is the profit in thousands of dollars, and x is the number of scissors sold in tens of thousands. What is the least number of scissors the company needs to sell to break even?

Cut Safe company manufactures scissors. The company uses the function P P (s)=−25s2+250s−400 (s)=-25 s^{2}+250 s-400 to model its profit, where P(x) P(x) is the profit in thousands of dollars, and x x is the number of scissors sold in tens of thousands. What is the least number of scissors the company needs to sell to break even?

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Q. Cut Safe company manufactures scissors. The company uses the function P P (s)=−25s2+250s−400 (s)=-25 s^{2}+250 s-400 to model its profit, where P(x) P(x) is the profit in thousands of dollars, and x x is the number of scissors sold in tens of thousands. What is the least number of scissors the company needs to sell to break even?
  1. Identify Break-Even Points: Identify the break-even points.\newlineThe break-even points occur when the profit P(s)P(s) is equal to zero.\newlineSo, we need to solve the equation −25s2+250s−400=0-25s^2 + 250s - 400 = 0 for ss.
  2. Factor Quadratic Equation: Factor the quadratic equation if possible.\newlineThe quadratic equation −25s2+250s−400=0-25s^2 + 250s - 400 = 0 can be factored by finding two numbers that multiply to −25×−400-25 \times -400 and add up to 250250. However, it's not immediately clear what those numbers are, so we might need to use the quadratic formula instead.
  3. Use Quadratic Formula: Use the quadratic formula to find the roots of the equation.\newlineThe quadratic formula is s=−b±b2−4ac2as = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}, where a=−25a = -25, b=250b = 250, and c=−400c = -400.
  4. Calculate Discriminant: Calculate the discriminant b2−4acb^2 - 4ac.\newlineDiscriminant = b2−4ac=(250)2−4(−25)(−400)=62500−40000=22500b^2 - 4ac = (250)^2 - 4(-25)(-400) = 62500 - 40000 = 22500.
  5. Calculate Square Root: Calculate the square root of the discriminant.\newline22500=150\sqrt{22500} = 150.
  6. Apply Quadratic Formula: Apply the quadratic formula to find the values of ss.s=−250±1502⋅−25s = \frac{{-250 \pm 150}}{{2 \cdot -25}}.
  7. Calculate Values of ss: Calculate the two possible values for ss.s1=−250+150−50=−100−50=2.s_1 = \frac{{-250 + 150}}{{-50}} = \frac{{-100}}{{-50}} = 2.s2=−250−150−50=−400−50=8.s_2 = \frac{{-250 - 150}}{{-50}} = \frac{{-400}}{{-50}} = 8.
  8. Determine Least Scissors: Determine the least number of scissors to break even.\newlineSince we are looking for the least number of scissors, we choose the smaller positive root, which is s=2s = 2.\newlineHowever, since ss represents tens of thousands of scissors, we need to multiply by 10,00010,000 to find the actual number of scissors.
  9. Calculate Least Scissors: Calculate the least number of scissors to break even in actual numbers.\newlineLeast number of scissors = s×10,000=2×10,000=20,000s \times 10,000 = 2 \times 10,000 = 20,000.

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