Discriminant Analysis

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Discriminant Analysis · Grade 9 · Worksheet 3

1. Tane is designing a parabolic skateboard ramp for a community park. The height of the ramp (in meters) above the ground at a horizontal distance x meters from the start is modeled by the quadratic function h(x) = -2x² + 24x + 18. The park committee requires that the ramp must reach a height of exactly 90 meters at some point to qualify for a national competition. Using the discriminant of the quadratic formula, determine whether this height is achievable. Answer: Yes Solution: Set h(x) = 90.
   Full step-by-step solution

   Step 1: Set h(x) = 90. -2x² + 24x + 18 = 90 Step 2: Rearrange to standard form: -2x² + 24x + 18 - 90 = 0 → -2x² + 24x - 72 = 0 Step 3: Identify coefficients: a = -2, b = 24, c = -72 Step 4: Calculate discriminant: D = b² - 4ac = 24² - 4(-2)(-72) = 576 - 4(144) = 576 - 576 = 0 Step 5: Interpret: Since D = 0, there is exactly one real solution, so the height of 90 meters is achievable. The answer is Yes.

2. 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.

3. Noah is testing a new water balloon launcher. The height of a water balloon, in meters, after t seconds is modeled by the quadratic function h(t) = -5t² + 28t + 8. Noah wants to know if the water balloon will ever reach a height of exactly 50 meters during its flight. Using the discriminant of the quadratic equation, determine whether this height is achievable. Answer: No Solution: Set up the equation h(t) = 50. -5t² + 28t + 8 = 50 Bring all terms to one side to get standard form. -5t² + 28t + 8 - 50 = 0 -5t² + 28t - 42 = 0 Identify a, b, and c.
   Full step-by-step solution

   Step 1: Set up the equation h(t) = 50. -5t² + 28t + 8 = 50 Step 2: Bring all terms to one side to get standard form. -5t² + 28t + 8 - 50 = 0 -5t² + 28t - 42 = 0 Step 3: Identify a, b, and c. a = -5, b = 28, c = -42 Step 4: Calculate the discriminant D = b² - 4ac. D = (28)² - 4(-5)(-42) D = 784 - 4(210) D = 784 - 840 D = -56 Step 5: Interpret the discriminant. Since D = -56 is less than 0, the quadratic equation has no real solutions. Step 6: Conclusion. Because there are no real values of t that make the height exactly 50 meters, the water balloon never reaches that height. The answer is No.

4. 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.

5. 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.

6. Noah is designing a water fountain for a park. The path of a water jet is modeled by the quadratic function h(x) = -2x² + 28x + 30, where h is the height in centimeters above the basin and x is the horizontal distance in centimeters from the nozzle. Noah wants to install a decorative arch that will pass exactly through the highest point of the water jet. Using the discriminant, determine whether the water jet ever reaches a height of 130 centimeters above the basin. Answer: No, the water jet never reaches a height of 130 centimeters. Solution: Set up the equation h(x) = 130. -2x² + 28x + 30 = 130 Bring all terms to one side: -2x² + 28x + 30 - 130 = 0, which simplifies to -2x² + 28x - 100 = 0.
   Full step-by-step solution

   Step 1: Set up the equation h(x) = 130. -2x² + 28x + 30 = 130 Step 2: Bring all terms to one side: -2x² + 28x + 30 - 130 = 0, which simplifies to -2x² + 28x - 100 = 0. Step 3: Identify coefficients for the discriminant formula: a = -2, b = 28, c = -100. Step 4: Calculate the discriminant D = b² - 4ac. D = (28)² - 4(-2)(-100) = 784 - 800 = -16. Step 5: Interpret the discriminant. Since D = -16 is less than 0, the quadratic equation has no real solutions. Step 6: Conclusion: There are no real values of x for which h(x) = 130. Therefore, the water jet never reaches a height of 130 centimeters.

7. Isabella sketches the graph of the quadratic function f(x) = 7x² + 28x + k on a coordinate plane. The parabola opens upward and its vertex lies exactly on the x-axis. What is the value of k? Answer: 28 Solution: The function is f(x) = 7x² + 28x + k. For the vertex to lie exactly on the x-axis, the parabola must have exactly one x-intercept.
   Full step-by-step solution

   Step 1: The function is f(x) = 7x² + 28x + k. For the vertex to lie exactly on the x-axis, the parabola must have exactly one x-intercept. This means the quadratic equation 7x² + 28x + k = 0 has exactly one real solution, which occurs when the discriminant is 0. Step 2: Identify the coefficients from the standard form ax² + bx + c = 0: a = 7, b = 28, c = k. Step 3: Write the discriminant formula: D = b² - 4ac. Step 4: Substitute the values: D = (28)² - 4 * 7 * k D = 784 - 28k Step 5: Set the discriminant equal to 0 (condition for exactly one solution): 784 - 28k = 0 Step 6: Solve for k: 28k = 784 k = 784 / 28 k = 28 Final answer: The value of k is 28.