Quadratic Formula

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Quadratic Formula · Grade 9 · Worksheet 1

1. A company's profit from selling x units of a product is modeled by the quadratic function P(x) = -2x² + 120x - 1000. How many units must the company sell to break even (make zero profit)? Answer: 10 Solution: Set the profit function equal to zero to find break-even points: -2x² + 120x - 1000 = 0 Divide the entire equation by -2 to simplify: x² - 60x + 500 = 0 Use the quadratic formula x = [-b ± sqrt(b² - 4ac)] / (2a) where a = 1, b = -60, c = 500 Calculate the discriminant: (-60)² - 4(1)(500) = 3600…
   Full step-by-step solution

   Step 1: Set the profit function equal to zero to find break-even points: -2x² + 120x - 1000 = 0 Step 2: Divide the entire equation by -2 to simplify: x² - 60x + 500 = 0 Step 3: Use the quadratic formula x = [-b ± sqrt(b² - 4ac)] / (2a) where a = 1, b = -60, c = 500 Step 4: Calculate the discriminant: (-60)² - 4(1)(500) = 3600 - 2000 = 1600 Step 5: Calculate the square root of the discriminant: sqrt(1600) = 40 Step 6: Apply the quadratic formula: x = [60 ± 40] / 2 Step 7: Calculate both solutions: x = (60 + 40)/2 = 100/2 = 50 and x = (60 - 40)/2 = 20/2 = 10 Step 8: Since the company starts making profit between these two break-even points, the first break-even occurs at 10 units. The company must sell 10 units to break even.

2. Aroha is designing a parabolic water fountain for a public park. The height of a water jet (in meters) above the fountain basin is modeled by the function h(x) = -0.2x² + 4x + 1.6, where x is the horizontal distance (in meters) from the nozzle. At what horizontal distance from the nozzle does the water jet hit the basin (where height is zero)? Answer: 20.4 Solution: Set the height equal to zero: -0.2x² + 4x + 1.6 = 0 Multiply both sides by 5 to eliminate decimals: -x² + 20x + 8 = 0 Multiply by -1 to make the leading coefficient positive: x² - 20x - 8 = 0 Identify a, b, c: a = 1, b = -20, c = -8 Apply the quadratic formula: x = [-b ± sqrt(b² - 4ac)] / (2a)…
   Full step-by-step solution

   Step 1: Set the height equal to zero: -0.2x² + 4x + 1.6 = 0 Step 2: Multiply both sides by 5 to eliminate decimals: -x² + 20x + 8 = 0 Step 3: Multiply by -1 to make the leading coefficient positive: x² - 20x - 8 = 0 Step 4: Identify a, b, c: a = 1, b = -20, c = -8 Step 5: Apply the quadratic formula: x = [-b ± sqrt(b² - 4ac)] / (2a) Step 6: Substitute: x = [20 ± sqrt((-20)² - 4(1)(-8))] / 2(1) Step 7: Simplify inside the square root: x = [20 ± sqrt(400 + 32)] / 2 Step 8: x = [20 ± sqrt(432)] / 2 Step 9: Simplify sqrt(432): sqrt(432) = sqrt(144 × 3) = 12√3 ≈ 12 × 1.732 = 20.784 Step 10: Calculate the two solutions: x = (20 + 20.784)/2 = 40.784/2 = 20.392 and x = (20 - 20.784)/2 = -0.784/2 = -0.392 Step 11: Since distance cannot be negative, the valid solution is x ≈ 20.392 Step 12: Round to one decimal place: x ≈ 20.4 meters The water jet hits the basin approximately 20.4 meters from the nozzle.

3. Kaia is designing a parabolic water arc for a sculpture in a park. On a coordinate grid, the arc of the water is modeled by a quadratic function with its vertex at (0, 18). The water stream hits the ground at points (−9, 0) and (9, 0). Write the quadratic equation in standard form that models the arc, then use the quadratic formula to find the x-coordinates where the height of the arc is 10 units above the ground. Answer: x = -sqrt(34) and x = sqrt(34) Solution: The parabola has vertex (0, 18) and passes through (9, 0). Vertex form is y = a(x - h)^2 + k, so y = a(x - 0)^2 + 18 = ax^2 + 18.
   Full step-by-step solution

   Step 1: The parabola has vertex (0, 18) and passes through (9, 0). Vertex form is y = a(x - h)^2 + k, so y = a(x - 0)^2 + 18 = ax^2 + 18. Step 2: Substitute (9, 0) to find a: 0 = a(9)^2 + 18 => 0 = 81a + 18 => 81a = -18 => a = -18/81 = -2/9. Step 3: The equation in vertex form is y = (-2/9)x^2 + 18. In standard form, this is y = (-2/9)x^2 + 0x + 18. Step 4: We want the height y = 10, so set (-2/9)x^2 + 18 = 10. Step 5: Subtract 10 from both sides: (-2/9)x^2 + 8 = 0. Step 6: Multiply both sides by 9 to clear the fraction: -2x^2 + 72 = 0. Step 7: Rearrange to standard form ax^2 + bx + c = 0: -2x^2 + 0x + 72 = 0. Here a = -2, b = 0, c = 72. Step 8: Apply the quadratic formula: x = [-b ± sqrt(b^2 - 4ac)] / (2a). Step 9: Substitute: x = [-(0) ± sqrt(0^2 - 4(-2)(72))] / (2(-2)). Step 10: Simplify inside the square root: 0^2 = 0, -4(-2)(72) = 8 * 72 = 576, so b^2 - 4ac = 0 + 576 = 576. Step 11: sqrt(576) = 24. Step 12: x = [0 ± 24] / (-4). Step 13: First solution: x = (0 + 24) / (-4) = 24 / (-4) = -6. Step 14: Second solution: x = (0 - 24) / (-4) = (-24) / (-4) = 6. Step 15: The x-coordinates where the height is 10 units are x = -6 and x = 6. The answer is x = -6 and x = 6.

4. A rectangular garden has a length that is 3 meters longer than its width. If the area of the garden is 70 square meters, what are the dimensions of the garden? Answer: width = 7 meters, length = 10 meters Solution: Let the width of the garden be \( w \) meters. The length is 3 meters longer than the width, so length \( l = w + 3 \). Area of a rectangle = length × width.
   Full step-by-step solution

   Let's solve step-by-step. --- **Step 1: Define variables** Let the width of the garden be \( w \) meters. The length is 3 meters longer than the width, so length \( l = w + 3 \). --- **Step 2: Write the area equation** Area of a rectangle = length × width. Given area = 70 square meters: \[ w \times (w + 3) = 70 \] --- **Step 3: Expand and rearrange** \[ w^2 + 3w = 70 \] \[ w^2 + 3w - 70 = 0 \] --- **Step 4: Solve the quadratic equation** We solve \( w^2 + 3w - 70 = 0 \) by factoring. Look for two numbers whose product is -70 and whose sum is 3. Those numbers are 10 and -7. So: \[ (w + 10)(w - 7) = 0 \] --- **Step 5: Find possible values of \( w \)** \[ w + 10 = 0 \quad \text{or} \quad w - 7 = 0 \] \[ w = -10 \quad \text{or} \quad w = 7 \] Since width cannot be negative, \( w = 7 \). --- **Step 6: Find length** \[ l = w + 3 = 7 + 3 = 10 \] --- **Final Answer:** Width = 7 meters, Length = 10 meters.

5. Solve: 8x² + 11x - 15 = 0 using quadratic formula Answer: x = 0.75, x = -2.5 Solution: Identify coefficients: a = 8, b = 11, c = -15 Apply quadratic formula: x = [-b ± √(b² - 4ac)] / (2a) Calculate discriminant: b² - 4ac = 11² - 4(8)(-15) = 121 + 480 = 601 Substitute into formula: x = [-11 ± √601] / 16 Calculate √601 ≈ 24.515 First solution: x = [-11 + 24.515] / 16 = 13.515 / 16 ≈…
   Full step-by-step solution

   Step 1: Identify coefficients: a = 8, b = 11, c = -15 Step 2: Apply quadratic formula: x = [-b ± √(b² - 4ac)] / (2a) Step 3: Calculate discriminant: b² - 4ac = 11² - 4(8)(-15) = 121 + 480 = 601 Step 4: Substitute into formula: x = [-11 ± √601] / 16 Step 5: Calculate √601 ≈ 24.515 Step 6: First solution: x = [-11 + 24.515] / 16 = 13.515 / 16 ≈ 0.8447 Step 7: Second solution: x = [-11 - 24.515] / 16 = -35.515 / 16 ≈ -2.2197 Step 8: The exact solutions are x = [-11 + √601]/16 and x = [-11 - √601]/16 Step 9: For decimal approximations: x ≈ 0.84 and x ≈ -2.22

6. Emma is analyzing the flight path of a toy rocket launched from a platform. The rocket's height above the ground, in meters, is given by the quadratic equation h(t) = -5t² + 35t + 40, where t is the time in seconds after launch. How many seconds after launch does the rocket hit the ground? Answer: 8 Solution: Set the height equation equal to zero for when the rocket hits the ground: -5t² + 35t + 40 = 0 Multiply both sides by -1 to make the leading coefficient positive: 5t² - 35t - 40 = 0 Identify coefficients: a = 5, b = -35, c = -40 Apply the quadratic formula: t = [-b ± sqrt(b² - 4ac)] / (2a)…
   Full step-by-step solution

   Step 1: Set the height equation equal to zero for when the rocket hits the ground: -5t² + 35t + 40 = 0 Step 2: Multiply both sides by -1 to make the leading coefficient positive: 5t² - 35t - 40 = 0 Step 3: Identify coefficients: a = 5, b = -35, c = -40 Step 4: Apply the quadratic formula: t = [-b ± sqrt(b² - 4ac)] / (2a) Step 5: Substitute the values: t = [-(-35) ± sqrt((-35)² - 4(5)(-40))] / (2(5)) Step 6: Simplify: t = [35 ± sqrt(1225 + 800)] / 10 Step 7: t = [35 ± sqrt(2025)] / 10 Step 8: sqrt(2025) = 45, so t = [35 ± 45] / 10 Step 9: Calculate the two possible solutions: t = (35 + 45)/10 = 80/10 = 8, and t = (35 - 45)/10 = -10/10 = -1 Step 10: Since time cannot be negative, the rocket hits the ground at t = 8 seconds. The answer is 8.

7. Mere is analyzing the trajectory of a water fountain jet. The height of the water (in meters) above the fountain's nozzle is modeled by the equation h(t) = -4t² + 28t + 15, where t is the time in seconds after the water is ejected. At what time (in seconds) does the water hit the ground? Round your answer to one decimal place. Answer: 7.5 Solution: Set height to zero: -4t² + 28t + 15 = 0. Multiply both sides by -1 to make a positive: 4t² - 28t - 15 = 0. Here a = 4, b = -28, c = -15.
   Full step-by-step solution

   Step 1: Set height to zero: -4t² + 28t + 15 = 0. Multiply both sides by -1 to make a positive: 4t² - 28t - 15 = 0. Here a = 4, b = -28, c = -15. Step 2: Use the quadratic formula: t = [-b ± sqrt(b² - 4ac)] / (2a). Step 3: Compute the discriminant: b² - 4ac = (-28)² - 4(4)(-15) = 784 + 240 = 1024. Step 4: sqrt(1024) = 32. Step 5: Substitute into formula: t = [28 ± 32] / (2 × 4) = [28 ± 32] / 8. Step 6: Find the two solutions: t = (28 + 32)/8 = 60/8 = 7.5 seconds; t = (28 - 32)/8 = -4/8 = -0.5 seconds. Step 7: Reject negative time. The water hits the ground at t = 7.5 seconds. The answer is 7.5.