Discriminant Analysis

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Discriminant Analysis · Grade 9 · Worksheet 2

1. For the quadratic equation 14x² - 35x + 21 = 0, calculate the discriminant and determine how many real solutions exist. Answer: 2 Solution: Identify the coefficients: a = 14, b = -35, c = 21 Calculate the discriminant: D = b² - 4ac = (-35)² - 4(14)(21) = 1225 - 1176 = 49 Analyze the discriminant: Since D = 49 and 49 > 0, the quadratic equation has two distinct real solutions.
   Full step-by-step solution

   Step 1: Identify the coefficients: a = 14, b = -35, c = 21 Step 2: Calculate the discriminant: D = b² - 4ac = (-35)² - 4(14)(21) = 1225 - 1176 = 49 Step 3: Analyze the discriminant: Since D = 49 and 49 > 0, the quadratic equation has two distinct real solutions. The answer is 2.

2. Mere sketches the graph of the quadratic function y = 3x² + 12x + k on a coordinate plane. The parabola opens upward and its vertex lies exactly on the x-axis. Determine the value of k and state how many real x-intercepts the graph has. Answer: k = 12, 1 real x-intercept Solution: Identify the coefficients from the quadratic equation y = 3x² + 12x + k. Here a = 3, b = 12, c = k. The vertex lies on the x-axis, meaning the parabola touches the x-axis at exactly one point.
   Full step-by-step solution

   Step 1: Identify the coefficients from the quadratic equation y = 3x² + 12x + k. Here a = 3, b = 12, c = k. Step 2: The vertex lies on the x-axis, meaning the parabola touches the x-axis at exactly one point. This happens when the discriminant is equal to 0. Step 3: Write the discriminant formula: D = b² - 4ac. Step 4: Substitute the known values: D = (12)² - 4(3)(k) = 144 - 12k. Step 5: Set the discriminant equal to 0 because there is exactly one x-intercept: 144 - 12k = 0. Step 6: Solve for k: 12k = 144, so k = 12. Step 7: Since D = 0, the quadratic has exactly one real x-intercept. The answer is k = 12, and the graph has 1 real x-intercept.

3. A company's profit from selling x units of a product is modeled by the quadratic function P(x) = -2x² + 120x - 1000. The company needs to determine how many units they must sell to break even (profit equals zero). Using the quadratic formula, find the number of units that represents the break-even point. Answer: 10 Solution: P(x) = -2x² + 120x - 1000 Break-even means profit = 0, so: -2x² + 120x - 1000 = 0 We can divide the entire equation by -2 to make it easier: (-2x²)/(-2) + (120x)/(-2) - (1000)/(-2) = 0 x² - 60x + 500 = 0 For equation ax² + bx + c = 0, a = 1, b = -60, c = 500 x = [ -b ± sqrt(b² - 4ac) ] / (2a) b²…
   Full step-by-step solution

   We are given the profit function: P(x) = -2x² + 120x - 1000 Break-even means profit = 0, so: -2x² + 120x - 1000 = 0 --- **Step 1: Simplify the equation** We can divide the entire equation by -2 to make it easier: (-2x²)/(-2) + (120x)/(-2) - (1000)/(-2) = 0 x² - 60x + 500 = 0 --- **Step 2: Identify coefficients for the quadratic formula** For equation ax² + bx + c = 0, a = 1, b = -60, c = 500 Quadratic formula: x = [ -b ± sqrt(b² - 4ac) ] / (2a) --- **Step 3: Calculate discriminant** b² - 4ac = (-60)² - 4(1)(500) = 3600 - 2000 = 1600 --- **Step 4: Apply quadratic formula** x = [ 60 ± sqrt(1600) ] / (2 * 1) sqrt(1600) = 40 So: x = (60 + 40)/2 = 100/2 = 50 x = (60 - 40)/2 = 20/2 = 10 --- **Step 5: Interpret the two solutions** The quadratic opens downward (since original a = -2), so profit is positive between the two break-even points. The smaller break-even point is x = 10 units, meaning the company first breaks even at 10 units sold. --- **Final answer:** 10

4. A company's profit from selling x units of a product is modeled by the quadratic function P(x) = -3x² + 150x - 1800. The company wants to know if they can achieve a profit of exactly $1000. Using the discriminant of the quadratic formula, determine whether this profit level is possible. Answer: A. no Solution: Set the profit function equal to 1000: -3x² + 150x - 1800 = 1000 Rearrange to standard quadratic form: -3x² + 150x - 2800 = 0 Multiply through by -1 to make calculations easier: 3x² - 150x + 2800 = 0 Identify coefficients: a = 3, b = -150, c = 2800 Calculate the discriminant: D = b² - 4ac =…
   Full step-by-step solution

   Step 1: Set the profit function equal to 1000: -3x² + 150x - 1800 = 1000 Step 2: Rearrange to standard quadratic form: -3x² + 150x - 2800 = 0 Step 3: Multiply through by -1 to make calculations easier: 3x² - 150x + 2800 = 0 Step 4: Identify coefficients: a = 3, b = -150, c = 2800 Step 5: Calculate the discriminant: D = b² - 4ac = (-150)² - 4(3)(2800) = 22500 - 33600 = -11100 Step 6: Since the discriminant is negative (-11100 < 0), there are no real solutions Step 7: This means the company cannot achieve exactly $1000 profit with this business model. The answer is no.

5. For the quadratic equation 4x² + 12x + 9 = 0, calculate the discriminant and determine how many real solutions exist. Answer: 1 Solution: Identify coefficients: a = 4, b = 12, c = 9 Calculate discriminant: D = b² - 4ac = 12² - 4(4)(9) = 144 - 144 = 0 Analyze discriminant: When D = 0, there is exactly one real solution (a repeated root) The answer is 1 real solution.
   Full step-by-step solution

   Step 1: Identify coefficients: a = 4, b = 12, c = 9 Step 2: Calculate discriminant: D = b² - 4ac = 12² - 4(4)(9) = 144 - 144 = 0 Step 3: Analyze discriminant: When D = 0, there is exactly one real solution (a repeated root) The answer is 1 real solution.

6. 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.

7. A quadratic function f(x) = 2x² - 8x + k has exactly one real root. What is the value of k? Answer: 8 Solution: A quadratic function has exactly one real root when its discriminant is equal to zero. Identify the coefficients. The quadratic function is f(x) = 2x² - 8x + k.
   Full step-by-step solution

   A quadratic function has exactly one real root when its discriminant is equal to zero. Step 1: Identify the coefficients. The quadratic function is f(x) = 2x² - 8x + k. So, a = 2, b = -8, c = k. Step 2: Write the formula for the discriminant. The discriminant D of a quadratic equation ax² + bx + c = 0 is given by D = b² - 4ac. Step 3: Substitute the coefficients into the discriminant formula. D = (-8)² - 4 * 2 * k D = 64 - 8k Step 4: Set the discriminant equal to zero for exactly one real root. For exactly one real root, D = 0. So, 64 - 8k = 0 Step 5: Solve for k. 64 - 8k = 0 Add 8k to both sides: 64 = 8k Divide both sides by 8: k = 64 / 8 k = 8 Therefore, the value of k is 8.

8. Aroha is designing a parabolic water fountain for a public park. The height of the water stream (in meters) is modeled by the quadratic function h(x) = -2x² + 16x + 10, where x is the horizontal distance from the nozzle in meters. The park's safety guidelines require that the water stream must reach a height of exactly 40 meters to be visible above a nearby hedge. Using the discriminant of the quadratic formula, determine whether the water stream reaches this required height. Answer: No, it does not reach exactly 40 meters. Solution: Set the height function equal to 40 meters: -2x² + 16x + 10 = 40 -2x² + 16x + 10 - 40 = 0 -2x² + 16x - 30 = 0 a = -2, b = 16, c = -30 Calculate the discriminant D = b² - 4ac: D = (16)² - 4(-2)(-30) D = 256 - 4(-2)(-30) D = 256 - (4 * 60) D = 256 - 240 D = 16 Since D = 16 > 0, the quadratic…
   Full step-by-step solution

   Step 1: Set the height function equal to 40 meters: -2x² + 16x + 10 = 40 Step 2: Rearrange to standard quadratic form: -2x² + 16x + 10 - 40 = 0 -2x² + 16x - 30 = 0 Step 3: Identify coefficients a, b, and c: a = -2, b = 16, c = -30 Step 4: Calculate the discriminant D = b² - 4ac: D = (16)² - 4(-2)(-30) D = 256 - 4(-2)(-30) D = 256 - (4 * 60) D = 256 - 240 D = 16 Step 5: Interpret the discriminant: Since D = 16 > 0, the quadratic equation has two distinct real solutions. This means there are two horizontal distances where the water stream is exactly 40 meters high. Step 6: Conclusion: Yes, the water stream does reach exactly 40 meters (at two different points).