Discriminant Analysis

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Discriminant Analysis · Grade 9 · Worksheet 1

1. A company's revenue from selling x units of a new gadget is modeled by the quadratic function R(x) = -3x² + 150x + 2000. The marketing team wants to know if they can achieve a revenue of exactly $5000. Using the discriminant of the quadratic formula, determine whether this revenue level is possible. Answer: A. yes Solution: Set up the equation for revenue equal to $5000: -3x² + 150x + 2000 = 5000 Rearrange to standard quadratic form: -3x² + 150x + 2000 - 5000 = 0 → -3x² + 150x - 3000 = 0 Multiply through by -1 to simplify: 3x² - 150x + 3000 = 0 Calculate the discriminant: D = b² - 4ac = (-150)² - 4(3)(3000) = 22500…
   Full step-by-step solution

   Step 1: Set up the equation for revenue equal to $5000: -3x² + 150x + 2000 = 5000 Step 2: Rearrange to standard quadratic form: -3x² + 150x + 2000 - 5000 = 0 → -3x² + 150x - 3000 = 0 Step 3: Multiply through by -1 to simplify: 3x² - 150x + 3000 = 0 Step 4: Calculate the discriminant: D = b² - 4ac = (-150)² - 4(3)(3000) = 22500 - 36000 = -13500 Step 5: Since the discriminant is negative (-13500 < 0), there are no real solutions. Step 6: Therefore, the company cannot achieve exactly $5000 in revenue with this business model.

2. A company's profit from selling x units of a product is modeled by the quadratic function P(x) = -2x² + 120x - 1000. The company wants to determine if they can achieve a profit of exactly $500. Using the discriminant of the quadratic formula, determine whether this profit level is possible and explain what this means about the number of units they would need to sell. Answer: No, it is not possible to achieve exactly $500 profit because the discriminant is negative. Solution: In quadratic profit modeling, the discriminant reveals whether specific profit targets can be achieved.
   Full step-by-step solution

   In quadratic profit modeling, the discriminant reveals whether specific profit targets can be achieved. A negative discriminant means the quadratic equation has no real solutions, indicating that particular profit level cannot be reached regardless of production quantity. This analysis helps businesses understand the range of possible profits given their cost and revenue structure.

3. A company's revenue from selling x units of a product is modeled by the quadratic function R(x) = -3x² + 150x - 1800. The company wants to determine if they can achieve a revenue of exactly $1000. Using the discriminant of the quadratic formula, determine whether this revenue level is possible. Answer: B. yes Solution: The discriminant of a quadratic equation ax² + bx + c = 0 is calculated as b² - 4ac. If the discriminant is positive, there are two real solutions; if zero, one real solution; if negative, no real solutions.
   Full step-by-step solution

   The discriminant of a quadratic equation ax² + bx + c = 0 is calculated as b² - 4ac. If the discriminant is positive, there are two real solutions; if zero, one real solution; if negative, no real solutions. In business applications, this tells us whether certain profit or revenue targets are achievable.

4. A soccer player kicks a ball from ground level with an initial upward velocity of 24 m/s. The height of the ball above the ground is modeled by the equation h(t) = -5t² + 24t, where t is time in seconds. The coach wants to know if the ball will reach a height of exactly 30 meters during its flight. Using the discriminant of the quadratic formula, determine whether this height is achievable. Answer: B. no Solution: Set up the equation: -5t² + 24t = 30 Rearrange to standard quadratic form: -5t² + 24t - 30 = 0 Multiply through by -1 to make calculations easier: 5t² - 24t + 30 = 0 Identify coefficients: a = 5, b = -24, c = 30 Calculate the discriminant: D = b² - 4ac = (-24)² - 4(5)(30) = 576 - 600 = -24 Since…
   Full step-by-step solution

   Step 1: Set up the equation: -5t² + 24t = 30 Step 2: Rearrange to standard quadratic form: -5t² + 24t - 30 = 0 Step 3: Multiply through by -1 to make calculations easier: 5t² - 24t + 30 = 0 Step 4: Identify coefficients: a = 5, b = -24, c = 30 Step 5: Calculate the discriminant: D = b² - 4ac = (-24)² - 4(5)(30) = 576 - 600 = -24 Step 6: Since the discriminant is negative (-24 < 0), there are no real solutions Step 7: This means the ball never reaches exactly 30 meters in height The answer is no.

5. Emma graphs the parabola of the quadratic function f(x) = 5x² - 20x + k. She observes that the parabola just touches the x-axis at exactly one point. What is the value of k? Answer: 20 Solution: The function is f(x) = 5x² - 20x + k. For a parabola to have exactly one x-intercept (touch the x-axis at one point), the quadratic equation 5x² - 20x + k = 0 must have exactly one real solution.
   Full step-by-step solution

   Step 1: The function is f(x) = 5x² - 20x + k. For a parabola to have exactly one x-intercept (touch the x-axis at one point), the quadratic equation 5x² - 20x + k = 0 must have exactly one real solution. This occurs when the discriminant is equal to 0. Step 2: Identify the coefficients from the standard form ax² + bx + c = 0: a = 5, b = -20, c = k. Step 3: Write the discriminant formula: D = b² - 4ac. Step 4: Substitute the values: D = (-20)² - 4 * 5 * k D = 400 - 20k Step 5: Set the discriminant equal to 0 (condition for exactly one solution): 400 - 20k = 0 Step 6: Solve for k: 20k = 400 k = 400 / 20 k = 20 Final answer: The value of k is 20.

6. Mere sketches the graph of the quadratic function y = 3x² + 12x + 13 on a coordinate plane. She notices the parabola does not cross the x-axis. Calculate the discriminant of this quadratic equation and confirm how many real x-intercepts the graph has. Answer: Discriminant is -12, and the graph has 0 real x-intercepts. Solution: Identify the coefficients from y = 3x² + 12x + 13. Here a = 3, b = 12, c = 13. Write the discriminant formula: D = b² - 4ac.
   Full step-by-step solution

   Step 1: Identify the coefficients from y = 3x² + 12x + 13. Here a = 3, b = 12, c = 13. Step 2: Write the discriminant formula: D = b² - 4ac. Step 3: Substitute the values: D = (12)² - 4 * 3 * 13. Step 4: Calculate b²: 12² = 144. Step 5: Calculate 4ac: 4 * 3 * 13 = 12 * 13 = 156. Step 6: Subtract: D = 144 - 156 = -12. Step 7: Interpret: Since D is negative (D < 0), the quadratic equation has no real solutions, meaning the parabola does not intersect the x-axis at any point. Final answer: Discriminant is -12, and the graph has 0 real x-intercepts.

7. 2x² + 7x + 3 = 0. Find discriminant. How many real solutions? Answer: 25, 2 Solution: Identify coefficients: a = 2, b = 7, c = 3 Calculate discriminant: D = b² - 4ac = 7² - 4(2)(3) = 49 - 24 = 25 Analyze discriminant: Since D = 25 > 0, there are two distinct real solutions The discriminant is 25 and there are 2 real solutions.
   Full step-by-step solution

   Step 1: Identify coefficients: a = 2, b = 7, c = 3 Step 2: Calculate discriminant: D = b² - 4ac = 7² - 4(2)(3) = 49 - 24 = 25 Step 3: Analyze discriminant: Since D = 25 > 0, there are two distinct real solutions The discriminant is 25 and there are 2 real solutions.