Discriminant Analysis

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Discriminant Analysis · Grade 9 · Worksheet 3

1. 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.

2. x² + 6x + 1 = 0 Answer: 32 Solution: Identify coefficients: a = 1, b = 6, c = 1 Calculate discriminant: D = b² - 4ac = 6² - 4(1)(1) = 36 - 4 = 32 Since D > 0, there are two distinct real roots The discriminant is 32.
   Full step-by-step solution

   Step 1: Identify coefficients: a = 1, b = 6, c = 1 Step 2: Calculate discriminant: D = b² - 4ac = 6² - 4(1)(1) = 36 - 4 = 32 Step 3: Since D > 0, there are two distinct real roots The discriminant is 32.

3. Emma sketches the graph of the quadratic function f(x) = 5x² - 20x + 15. The parabola opens upward. Calculate the discriminant of this quadratic equation and determine how many real x-intercepts the graph has. Answer: Discriminant is 100, and the graph has 2 real x-intercepts. Solution: Identify the coefficients from the standard form f(x) = ax² + bx + c. Here, a = 5, b = -20, c = 15. Write the discriminant formula: D = b² - 4ac.
   Full step-by-step solution

   Step 1: Identify the coefficients from the standard form f(x) = ax² + bx + c. Here, a = 5, b = -20, c = 15. Step 2: Write the discriminant formula: D = b² - 4ac. Step 3: Substitute the values: D = (-20)² - 4 * 5 * 15. Step 4: Calculate step by step: (-20)² = 400. Then 4 * 5 * 15 = 300. So D = 400 - 300 = 100. Step 5: Interpret the discriminant. Since D = 100 > 0, the quadratic equation has two distinct real roots. On a graph, these correspond to two distinct x-intercepts. Final answer: The discriminant is 100, and the graph has 2 real x-intercepts.

4. A company's profit from selling x units of a new smartphone accessory is modeled by the quadratic function P(x) = -4x² + 200x - 2400. The company wants to know if they can achieve a profit of exactly $800. Using the discriminant of the quadratic formula, determine whether this profit level is possible. Answer: A. yes Solution: Set up the equation P(x) = 800 -4x² + 200x - 2400 = 800 -4x² + 200x - 2400 - 800 = 0 -4x² + 200x - 3200 = 0 Identify coefficients for the discriminant formula For equation ax² + bx + c = 0: a = -4, b = 200, c = -3200 Calculate the discriminant D = b² - 4ac D = (200)² - 4(-4)(-3200) D = 40000 -…
   Full step-by-step solution

   Step 1: Set up the equation P(x) = 800 -4x² + 200x - 2400 = 800 Step 2: Bring all terms to one side -4x² + 200x - 2400 - 800 = 0 -4x² + 200x - 3200 = 0 Step 3: Identify coefficients for the discriminant formula For equation ax² + bx + c = 0: a = -4, b = 200, c = -3200 Step 4: Calculate the discriminant D = b² - 4ac D = (200)² - 4(-4)(-3200) D = 40000 - 4(-4)(-3200) D = 40000 - (16)(-3200) D = 40000 - (-51200) D = 40000 + 51200 D = 91200 Step 5: Interpret the discriminant Since D > 0, there are two distinct real solutions for x. Step 6: Conclusion Since the discriminant is positive, there are real values of x (production levels) where the profit equals exactly $800. Therefore, this profit level is possible.

5. A company's profit from selling x units of a product is modeled by the quadratic function P(x) = -2x² + 120x - 1600. The company needs to determine how many different quantities of units they could sell to break even (make zero profit). Using the discriminant of the quadratic formula, find how many distinct break-even points exist for this business model. Answer: 2 Solution: The profit function is P(x) = -2x² + 120x - 1600. Break-even means P(x) = 0. So we solve: -2x² + 120x - 1600 = 0.
   Full step-by-step solution

   Step 1: Understand the problem The profit function is P(x) = -2x² + 120x - 1600. Break-even means P(x) = 0. So we solve: -2x² + 120x - 1600 = 0. Step 2: Recall the discriminant in the quadratic formula For a quadratic equation ax² + bx + c = 0, the discriminant is D = b² - 4ac. If D > 0, there are 2 distinct real solutions. If D = 0, there is 1 real solution. If D < 0, there are no real solutions. Step 3: Identify a, b, c From -2x² + 120x - 1600 = 0, we have: a = -2 b = 120 c = -1600 Step 4: Calculate the discriminant D = b² - 4ac D = (120)² - 4(-2)(-1600) D = 14400 - 4(-2)(-1600) First, compute 4(-2)(-1600): 4 * (-2) = -8 -8 * (-1600) = 12800 So D = 14400 - 12800 D = 1600 Step 5: Interpret the discriminant D = 1600, which is greater than 0. Since D > 0, the quadratic equation has two distinct real solutions. Step 6: Conclusion The number of distinct break-even points (quantities x where profit is zero) is 2. Final answer: 2

6. Nova is analyzing the graph of a quadratic function f(x) = ax² + bx + c. The parabola opens downward and its vertex lies exactly on the x-axis. The equation is given as f(x) = -x² + 6x + c. What value of c makes the vertex lie on the x-axis, and what is the discriminant in that case? Answer: c = -9, discriminant = 0 Solution: Write the quadratic in standard form: f(x) = -x² + 6x + c. Here a = -1, b = 6, c = c. For the vertex to lie on the x-axis, the quadratic must have exactly one real x-intercept.
   Full step-by-step solution

   Step 1: Write the quadratic in standard form: f(x) = -x² + 6x + c. Here a = -1, b = 6, c = c. Step 2: For the vertex to lie on the x-axis, the quadratic must have exactly one real x-intercept. This means the discriminant D must be equal to 0. Step 3: The discriminant formula is D = b² - 4ac. Step 4: Substitute the known values: D = 6² - 4(-1)(c) = 36 + 4c. Step 5: Set D = 0 for one real solution: 36 + 4c = 0. Step 6: Solve for c: 4c = -36, so c = -9. Step 7: Now calculate the discriminant with c = -9: D = 36 + 4(-9) = 36 - 36 = 0. Final answer: c = -9 and the discriminant is 0.

7. Kaia is analyzing the graph of a quadratic function. The parabola opens downward and its equation is y = -3x² + 12x - 12. The vertex of the parabola lies directly on the x-axis. Calculate the discriminant of this quadratic equation and state how many real x-intercepts the graph has. Answer: Discriminant is 0, and the graph has 1 real x-intercept. Solution: Identify the coefficients from the equation y = -3x² + 12x - 12 in the standard form ax² + bx + c = 0. a = -3, b = 12, c = -12 Write the discriminant formula. D = b² - 4ac Substitute the values into the formula.
   Full step-by-step solution

   Step 1: Identify the coefficients from the equation y = -3x² + 12x - 12 in the standard form ax² + bx + c = 0. a = -3, b = 12, c = -12 Step 2: Write the discriminant formula. D = b² - 4ac Step 3: Substitute the values into the formula. D = (12)² - 4(-3)(-12) Step 4: Calculate step by step. First, 12² = 144. Then, 4(-3)(-12) = 4 * 36 = 144. So, D = 144 - 144 = 0. Step 5: Interpret the discriminant. If D > 0, there are two distinct real x-intercepts. If D = 0, there is exactly one real x-intercept (the vertex touches the x-axis). If D < 0, there are no real x-intercepts. Here, D = 0, so there is exactly one real x-intercept. Final answer: The discriminant is 0, and the graph has 1 real x-intercept.