Quadratic Formula

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Answer Key & Explanations

Quadratic Formula · Grade 9 · Worksheet 2

1. Emma is designing a skateboard ramp that forms a parabolic path. The ramp's height above ground follows the equation h(t) = -2t² + 12t, where h is the height in feet and t is the time in seconds after launch. At what time does the skateboard reach its maximum height? Answer: 3 Solution: The height function is h(t) = -2t² + 12t This is a quadratic function in standard form ax² + bx + c, where a = -2, b = 12, c = 0 For a quadratic function, the vertex occurs at t = -b/(2a) Substitute the values: t = -12/(2×(-2)) = -12/(-4) = 3 The skateboard reaches its maximum height at t = 3…
   Full step-by-step solution

   Step 1: The height function is h(t) = -2t² + 12t Step 2: This is a quadratic function in standard form ax² + bx + c, where a = -2, b = 12, c = 0 Step 3: For a quadratic function, the vertex occurs at t = -b/(2a) Step 4: Substitute the values: t = -12/(2×(-2)) = -12/(-4) = 3 Step 5: The skateboard reaches its maximum height at t = 3 seconds The answer is 3.

2. 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.

3. Solve: 7x² + 12x - 2 = 0 using quadratic formula Answer: x = (-6 + √50)/7, x = (-6 - √50)/7 Solution: Identify coefficients: a = 7, b = 12, c = -2 Apply quadratic formula: x = [-b ± √(b² - 4ac)] / (2a) Substitute values: x = [-12 ± √(12² - 4×7×(-2))] / (2×7) Calculate discriminant: 12² = 144, 4×7×(-2) = -56, so 144 - (-56) = 144 + 56 = 200 Simplify: x = [-12 ± √200] / 14 Factor numerator: x =…
   Full step-by-step solution

   Step 1: Identify coefficients: a = 7, b = 12, c = -2 Step 2: Apply quadratic formula: x = [-b ± √(b² - 4ac)] / (2a) Step 3: Substitute values: x = [-12 ± √(12² - 4×7×(-2))] / (2×7) Step 4: Calculate discriminant: 12² = 144, 4×7×(-2) = -56, so 144 - (-56) = 144 + 56 = 200 Step 5: Simplify: x = [-12 ± √200] / 14 Step 6: Factor numerator: x = [2(-6 ± √50)] / 14 Step 7: Simplify fraction: x = (-6 ± √50)/7 Step 8: Final answers: x = (-6 + √50)/7 and x = (-6 - √50)/7

4. A company's profit from selling x units of a product is modeled by the quadratic function P(x) = -2x² + 120x - 1000. The company wants to know how many units they need to sell to break even (make zero profit). How many units must be sold to break even? Answer: 10 and 50 Solution: To find when a company breaks even, we set the profit function equal to zero and solve the resulting quadratic equation. The quadratic formula can be used to find the roots of any quadratic equation in standard form.
   Full step-by-step solution

   To find when a company breaks even, we set the profit function equal to zero and solve the resulting quadratic equation. The quadratic formula can be used to find the roots of any quadratic equation in standard form. In business applications, quadratic functions often model revenue, cost, or profit scenarios where there are break-even points.

5. Solve: 5x² - 15x - 20 = 0 using quadratic formula Answer: x = 4, x = -1 Solution: Identify coefficients: a = 5, b = -15, c = -20 Calculate discriminant: b² - 4ac = (-15)² - 4(5)(-20) = 225 + 400 = 625 Apply quadratic formula: x = [15 ± √625] / (2×5) Simplify: x = [15 ± 25] / 10 Calculate first solution: x = (15 + 25)/10 = 40/10 = 4 Calculate second solution: x = (15 - 25)/10…
   Full step-by-step solution

   Step 1: Identify coefficients: a = 5, b = -15, c = -20 Step 2: Calculate discriminant: b² - 4ac = (-15)² - 4(5)(-20) = 225 + 400 = 625 Step 3: Apply quadratic formula: x = [15 ± √625] / (2×5) Step 4: Simplify: x = [15 ± 25] / 10 Step 5: Calculate first solution: x = (15 + 25)/10 = 40/10 = 4 Step 6: Calculate second solution: x = (15 - 25)/10 = (-10)/10 = -1 Step 7: Final solutions: x = 4 and x = -1

6. 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

7. Solve: 9x² - 13x - 7 = 0 using quadratic formula Answer: x = (13 ± √(169 + 252))/18 = (13 ± √421)/18 Solution: Identify coefficients: a = 9, b = -13, c = -7 Calculate discriminant: b² - 4ac = (-13)² - 4(9)(-7) = 169 - (-252) = 169 + 252 = 421 Apply quadratic formula: x = [13 ± √421] / (2 × 9) Simplify denominator: x = [13 ± √421] / 18 Write final answer: x = (13 + √421)/18 or x = (13 - √421)/18
   Full step-by-step solution

   Step 1: Identify coefficients: a = 9, b = -13, c = -7 Step 2: Calculate discriminant: b² - 4ac = (-13)² - 4(9)(-7) = 169 - (-252) = 169 + 252 = 421 Step 3: Apply quadratic formula: x = [13 ± √421] / (2 × 9) Step 4: Simplify denominator: x = [13 ± √421] / 18 Step 5: Write final answer: x = (13 + √421)/18 or x = (13 - √421)/18

8. 3x² - 11x + 6 = 0 Answer: x = 3, x = 2/3 Solution: Identify coefficients: a = 3, b = -11, c = 6 Calculate the discriminant: b² - 4ac = (-11)² - 4(3)(6) = 121 - 72 = 49 Apply quadratic formula: x = [11 ± √49] / (2×3) = [11 ± 7] / 6 Case 1: x = (11 + 7) / 6 = 18 / 6 = 3 Case 2: x = (11 - 7) / 6 = 4 / 6 = 2/3 The solutions are x = 3 and x = 2/3
   Full step-by-step solution

   Step 1: Identify coefficients: a = 3, b = -11, c = 6 Step 2: Calculate the discriminant: b² - 4ac = (-11)² - 4(3)(6) = 121 - 72 = 49 Step 3: Apply quadratic formula: x = [11 ± √49] / (2×3) = [11 ± 7] / 6 Step 4: Solve for both cases: Case 1: x = (11 + 7) / 6 = 18 / 6 = 3 Case 2: x = (11 - 7) / 6 = 4 / 6 = 2/3 Step 5: The solutions are x = 3 and x = 2/3

9. A rocket is launched from a platform 20 meters above ground level. Its height h (in meters) after t seconds is given by the equation h(t) = -5t² + 30t + 20. At what time does the rocket reach its maximum height? Answer: 3 Solution: The height function is h(t) = -5t² + 30t + 20 This is a quadratic function in the form at² + bt + c, where a = -5, b = 30, c = 20 Since a is negative, the parabola opens downward, so the vertex represents the maximum height The t-coordinate of the vertex is given by t = -b/(2a) Substitute the…
   Full step-by-step solution

   Step 1: The height function is h(t) = -5t² + 30t + 20 Step 2: This is a quadratic function in the form at² + bt + c, where a = -5, b = 30, c = 20 Step 3: Since a is negative, the parabola opens downward, so the vertex represents the maximum height Step 4: The t-coordinate of the vertex is given by t = -b/(2a) Step 5: Substitute the values: t = -30/(2×(-5)) = -30/(-10) = 3 Step 6: The rocket reaches its maximum height at t = 3 seconds The answer is 3.