Discriminant Analysis

lessonbunny.com

Answer Key & Explanations

Discriminant Analysis · Grade 9 · Worksheet 1

1. Emma is analyzing the trajectory of a drone she is testing. The height of the drone above the ground, in meters, is modeled by the quadratic function h(t) = -3t² + 21t + 27, where t is time in seconds. Emma wants to know if the drone will ever reach a height of exactly 75 meters during its flight. Using the discriminant of the quadratic formula, determine whether this height is achievable. Answer: No, the height of 75 meters is not achievable. Solution: Set h(t) = 75: -3t² + 21t + 27 = 75 Bring all terms to one side: -3t² + 21t + 27 - 75 = 0 => -3t² + 21t - 48 = 0 Multiply both sides by -1 to make the leading coefficient positive: 3t² - 21t + 48 = 0 Identify coefficients: a = 3, b = -21, c = 48 Calculate discriminant D = b² - 4ac: D = (-21)² -…
   Full step-by-step solution

   Step 1: Set h(t) = 75: -3t² + 21t + 27 = 75 Step 2: Bring all terms to one side: -3t² + 21t + 27 - 75 = 0 => -3t² + 21t - 48 = 0 Step 3: Multiply both sides by -1 to make the leading coefficient positive: 3t² - 21t + 48 = 0 Step 4: Identify coefficients: a = 3, b = -21, c = 48 Step 5: Calculate discriminant D = b² - 4ac: D = (-21)² - 4(3)(48) = 441 - 576 = -135 Step 6: Since D < 0, the quadratic equation has no real solutions. Step 7: Therefore, there is no real time t when the drone's height is exactly 75 meters. The height of 75 meters is not achievable.

2. Sophia is designing a parabolic arch for a new garden entrance. The shape of the arch can be modeled by the equation h(x) = -2x² + 28x, where h(x) represents the height in feet at a horizontal distance x feet from the left base. She wants to know if the arch will be tall enough to allow a 100-foot-tall sculpture to pass through the center. Using the discriminant of the quadratic equation, determine whether the arch ever reaches a height of exactly 100 feet. Answer: No, the arch does not reach a height of exactly 100 feet. Solution: Set the height equation equal to 100. -2x² + 28x = 100 Rearrange into standard quadratic form ax² + bx + c = 0. -2x² + 28x - 100 = 0 Identify the coefficients.
   Full step-by-step solution

   Step 1: Set the height equation equal to 100. -2x² + 28x = 100 Step 2: Rearrange into standard quadratic form ax² + bx + c = 0. -2x² + 28x - 100 = 0 Step 3: Identify the coefficients. a = -2, b = 28, c = -100 Step 4: Calculate the discriminant D = b² - 4ac. D = (28)² - 4(-2)(-100) D = 784 - 4(-2)(-100) D = 784 - (8)(-100) D = 784 - (-800) D = 784 - 800 D = -16 Step 5: Interpret the discriminant. Since D = -16 < 0, the quadratic equation has no real solutions. Step 6: Conclusion. Because there are no real solutions, there is no horizontal distance x where the arch reaches a height of exactly 100 feet. Therefore, the arch does not allow the sculpture to pass through at that height. The answer is no, the arch does not reach a height of exactly 100 feet.

3. Mason launches a model rocket from the ground with an initial upward velocity of 30 m/s. The height of the rocket above the ground, in meters, is modeled by the quadratic function h(t) = -5t² + 30t, where t is the time in seconds after launch. Isabella, his classmate, claims the rocket will reach a height of exactly 50 meters during its flight. Using the discriminant of the quadratic equation, determine whether Isabella is correct. Answer: no Solution: Set the height equal to 50 meters: -5t² + 30t = 50 Rearrange to standard quadratic form: -5t² + 30t - 50 = 0 Identify coefficients: a = -5, b = 30, c = -50 Calculate the discriminant D = b² - 4ac D = (30)² - 4(-5)(-50) D = 900 - 4(250) D = 900 - 1000 D = -100 Since D < 0, the quadratic equation…
   Full step-by-step solution

   Step 1: Set the height equal to 50 meters: -5t² + 30t = 50 Step 2: Rearrange to standard quadratic form: -5t² + 30t - 50 = 0 Step 3: Identify coefficients: a = -5, b = 30, c = -50 Step 4: Calculate the discriminant D = b² - 4ac D = (30)² - 4(-5)(-50) D = 900 - 4(250) D = 900 - 1000 D = -100 Step 5: Since D < 0, the quadratic equation has no real solutions. Step 6: Therefore, the rocket never reaches exactly 50 meters. Isabella is incorrect. The answer is no.

4. 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.

5. Aroha is a park ranger monitoring the flight of a hawk. The hawk's height above the ground, in meters, t seconds after it begins its dive is modeled by the quadratic function h(t) = -5t² + 36t + 12. Aroha needs to determine if the hawk will ever reach a height of exactly 80 meters during its dive. Using the discriminant of the quadratic formula, determine whether this height is achievable. Answer: No Solution: Set the height equal to 80. -5t² + 36t + 12 = 80 Rearrange to standard form (ax² + bx + c = 0). -5t² + 36t + 12 - 80 = 0 -5t² + 36t - 68 = 0 Identify the coefficients.
   Full step-by-step solution

   Step 1: Set the height equal to 80. -5t² + 36t + 12 = 80 Step 2: Rearrange to standard form (ax² + bx + c = 0). -5t² + 36t + 12 - 80 = 0 -5t² + 36t - 68 = 0 Step 3: Identify the coefficients. a = -5, b = 36, c = -68 Step 4: Calculate the discriminant, D = b² - 4ac. D = (36)² - 4(-5)(-68) D = 1296 - 4(-5)(-68) D = 1296 - (4 * 340) D = 1296 - 1360 D = -64 Step 5: Interpret the discriminant. Since D = -64 is less than 0, the quadratic equation has no real solutions. Conclusion: Because there are no real values of t that satisfy the equation, the hawk never reaches exactly 80 meters during its dive. The answer is No.

6. A quadratic function is defined as f(x) = 3x² - 12x + c. If the graph of this function intersects the x-axis at exactly one point, what is the value of c? Answer: 12 Solution: For a quadratic function to intersect the x-axis at exactly one point, the discriminant must equal zero. The discriminant formula is D = b² - 4ac, where a = 3, b = -12, and c is the unknown constant.
   Full step-by-step solution

   Step 1: For a quadratic function to intersect the x-axis at exactly one point, the discriminant must equal zero. Step 2: The discriminant formula is D = b² - 4ac, where a = 3, b = -12, and c is the unknown constant. Step 3: Set up the equation: (-12)² - 4(3)(c) = 0 Step 4: Simplify: 144 - 12c = 0 Step 5: Solve for c: 144 = 12c Step 6: Divide both sides by 12: c = 12 The answer is 12.