Discriminant Analysis

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Discriminant Analysis · Grade 9 · Worksheet 2

1. 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.

2. A drone is launched from a platform 20 meters high with an initial upward velocity of 15 m/s. The drone's height above ground is modeled by the equation h(t) = -5t² + 15t + 20, where t is time in seconds. Determine how many seconds it will take for the drone to reach the ground. Answer: 4 Solution: We are given the height equation: h(t) = -5t² + 15t + 20. The drone reaches the ground when h(t) = 0.
   Full step-by-step solution

   We are given the height equation: h(t) = -5t² + 15t + 20. The drone reaches the ground when h(t) = 0. Step 1: Set h(t) = 0 -5t² + 15t + 20 = 0 Step 2: Multiply the entire equation by -1 to make the leading coefficient positive 5t² - 15t - 20 = 0 Step 3: Divide the entire equation by 5 to simplify t² - 3t - 4 = 0 Step 4: Solve the quadratic equation by factoring We look for two numbers that multiply to -4 and add to -3. Those numbers are -4 and +1. So: (t - 4)(t + 1) = 0 Step 5: Find the roots t - 4 = 0 => t = 4 t + 1 = 0 => t = -1 Step 6: Interpret the results t = -1 is not meaningful because time cannot be negative. t = 4 is the time when the drone reaches the ground after launch. Final answer: 4 seconds.

3. A drone is launched from a platform 12 meters high. Its height above ground is modeled by the function h(t) = -2t² + 8t + 12, where t is time in seconds. The drone operator needs to know if the drone will clear a tree that is 20 meters tall. Using the discriminant, determine whether the drone reaches a height of at least 20 meters at any point during its flight. Answer: B. no Solution: In quadratic functions modeling projectile motion, the discriminant helps determine if a certain height is achievable.
   Full step-by-step solution

   In quadratic functions modeling projectile motion, the discriminant helps determine if a certain height is achievable. When you set the height function equal to a target value and rearrange to standard quadratic form, a positive discriminant indicates the object reaches that height at two different times, a zero discriminant means it just touches that height at one moment, and a negative discriminant means it never reaches that height. This analysis works for any quadratic height function and target value.

4. x² + 7x + 13 = 0. Calculate discriminant. How many real solutions? Answer: 0 Solution: Identify coefficients: a = 1, b = 7, c = 13 Calculate discriminant: D = b² - 4ac = 7² - 4(1)(13) = 49 - 52 = -3 Since D < 0, there are no real solutions The answer is 0 real solutions.
   Full step-by-step solution

   Step 1: Identify coefficients: a = 1, b = 7, c = 13 Step 2: Calculate discriminant: D = b² - 4ac = 7² - 4(1)(13) = 49 - 52 = -3 Step 3: Since D < 0, there are no real solutions The answer is 0 real solutions.

5. For the quadratic equation 4x² + 8x + 4 = 0, calculate the discriminant and determine how many real solutions exist. Answer: 1 Solution: Identify the coefficients: a = 4, b = 8, c = 4 Calculate the discriminant: D = b² - 4ac = 8² - 4(4)(4) = 64 - 64 = 0 Analyze the discriminant: When D = 0, the quadratic equation has exactly one real solution (a repeated root).
   Full step-by-step solution

   Step 1: Identify the coefficients: a = 4, b = 8, c = 4 Step 2: Calculate the discriminant: D = b² - 4ac = 8² - 4(4)(4) = 64 - 64 = 0 Step 3: Analyze the discriminant: When D = 0, the quadratic equation has exactly one real solution (a repeated root). The answer is 1.

6. A drone is launched from a platform and follows a parabolic path described by the equation h(t) = -5t² + 20t + 15, where h represents the height in meters and t represents time in seconds. The drone operator wants to know if the drone will clear a tree that is 25 meters tall. Using the discriminant of the quadratic equation, determine whether the drone reaches a height of 25 meters at any point during its flight. Answer: No, the drone does not reach 25 meters. Solution: In quadratic functions modeling projectile motion, the discriminant helps determine if a certain height is achievable.
   Full step-by-step solution

   In quadratic functions modeling projectile motion, the discriminant helps determine if a certain height is achievable. When you set the height function equal to a specific value and rearrange to form a quadratic equation, a positive discriminant indicates the object reaches that height at two different times, a zero discriminant means it just touches that height at one instant, and a negative discriminant means it never reaches that height. This analysis applies to various real-world scenarios like sports trajectories or engineering applications.

7. 2x² + 10x + 5 = 0 Answer: 2 Solution: Identify coefficients: a = 2, b = 10, c = 5 Calculate discriminant: D = b² - 4ac = 10² - 4(2)(5) = 100 - 40 = 60 Since D > 0 (60 > 0), there are two distinct real solutions The answer is 2.
   Full step-by-step solution

   Step 1: Identify coefficients: a = 2, b = 10, c = 5 Step 2: Calculate discriminant: D = b² - 4ac = 10² - 4(2)(5) = 100 - 40 = 60 Step 3: Since D > 0 (60 > 0), there are two distinct real solutions The answer is 2.