Quadratic Formula

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Answer Key & Explanations

Quadratic Formula · Grade 9 · Worksheet 1

1. Solve: 5x² - 14x + 8 = 0 using quadratic formula Answer: x = 2, x = 0.8 Solution: Identify coefficients: a = 5, b = -14, c = 8 Apply quadratic formula: x = [-b ± √(b²-4ac)] / 2a Calculate discriminant: b²-4ac = (-14)² - 4(5)(8) = 196 - 160 = 36 √36 = 6 First solution: x = [14 + 6] / 10 = 20/10 = 2 Second solution: x = [14 - 6] / 10 = 8/10 = 0.8 The solutions are x = 2 and x = 0.8
   Full step-by-step solution

   Step 1: Identify coefficients: a = 5, b = -14, c = 8 Step 2: Apply quadratic formula: x = [-b ± √(b²-4ac)] / 2a Step 3: Calculate discriminant: b²-4ac = (-14)² - 4(5)(8) = 196 - 160 = 36 Step 4: √36 = 6 Step 5: First solution: x = [14 + 6] / 10 = 20/10 = 2 Step 6: Second solution: x = [14 - 6] / 10 = 8/10 = 0.8 Step 7: The solutions are x = 2 and x = 0.8

2. 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 determine the number of units they need to sell to break even (make zero profit). How many units must they sell to break even? Answer: 10 and 50 Solution: In business applications, breaking even occurs when revenue equals costs, resulting in zero profit. Quadratic functions often model profit scenarios where there's an optimal production level.
   Full step-by-step solution

   In business applications, breaking even occurs when revenue equals costs, resulting in zero profit. Quadratic functions often model profit scenarios where there's an optimal production level. To find break-even points, we set the profit function equal to zero and solve the resulting quadratic equation. This concept applies to many real-world situations where there are both fixed and variable costs involved in production.

3. Ava is analyzing the trajectory of a water fountain in a park. The height of a water droplet (in meters) above the fountain's nozzle is modeled by the equation h(t) = -7t² + 91t + 42, where t is the time in seconds after the droplet leaves the nozzle. How many seconds after leaving the nozzle does the water droplet hit the ground? Round your answer to two decimal places. Answer: 13.44 Solution: Set the height equation to zero: -7t² + 91t + 42 = 0. Identify coefficients: a = -7, b = 91, c = 42. Substitute the values: t = [-91 ± sqrt(91² - 4(-7)(42))] / (2(-7)).
   Full step-by-step solution

   Step 1: Set the height equation to zero: -7t² + 91t + 42 = 0. Step 2: Identify coefficients: a = -7, b = 91, c = 42. Step 3: Apply the quadratic formula: t = [-b ± sqrt(b² - 4ac)] / (2a). Step 4: Substitute the values: t = [-91 ± sqrt(91² - 4(-7)(42))] / (2(-7)). Step 5: Calculate the discriminant: 91² = 8281, -4(-7)(42) = 1176, so discriminant = 8281 + 1176 = 9457. Step 6: Compute sqrt(9457) ≈ 97.25. Step 7: Substitute back: t = [-91 ± 97.25] / (-14). Step 8: Find the two solutions: t = [-91 + 97.25] / (-14) = 6.25 / (-14) ≈ -0.446 (not valid, time cannot be negative). t = [-91 - 97.25] / (-14) = -188.25 / (-14) ≈ 13.446. Step 9: Round to two decimal places: t ≈ 13.44 seconds. The water droplet hits the ground approximately 13.44 seconds after leaving the nozzle.

4. A right triangle is drawn on a coordinate plane with vertices at (0,0), (x,0), and (0,8). The hypotenuse has a length of 10 units. Using the Pythagorean theorem, determine the value of x and write the quadratic equation that models this situation. Solve the equation to find the possible values for x. Answer: 6 Solution: Step 1: Apply the Pythagorean theorem: a² + b² = c² Step 2: Substitute the known values: x² + 8² = 10² Step 3: Calculate the squares: x² + 64 = 100 Step 4: Subtract 64 from both sides: x² = 36 Step 5: Take the square root of both sides: x = ±6 Step 6: Since length cannot be negative in this…
   Full step-by-step solution

   Step 1: Apply the Pythagorean theorem: a² + b² = c² Step 2: Substitute the known values: x² + 8² = 10² Step 3: Calculate the squares: x² + 64 = 100 Step 4: Subtract 64 from both sides: x² = 36 Step 5: Take the square root of both sides: x = ±6 Step 6: Since length cannot be negative in this geometric context, x = 6 The answer is 6.

5. Solve: 7x² - 11x - 13 = 0 using quadratic formula Answer: x = (11 + √(11² - 4×7×(-13)))/(2×7) and x = (11 - √(11² - 4×7×(-13)))/(2×7) Solution: Identify coefficients: a = 7, b = -11, c = -13 Write quadratic formula: x = [-b ± √(b² - 4ac)] / 2a Substitute values: x = [11 ± √((-11)² - 4×7×(-13))] / (2×7) Calculate discriminant: (-11)² - 4×7×(-13) = 121 - (-364) = 121 + 364 = 485 Simplify: x = [11 ± √485] / 14 Final solutions: x = (11 +…
   Full step-by-step solution

   Step 1: Identify coefficients: a = 7, b = -11, c = -13 Step 2: Write quadratic formula: x = [-b ± √(b² - 4ac)] / 2a Step 3: Substitute values: x = [11 ± √((-11)² - 4×7×(-13))] / (2×7) Step 4: Calculate discriminant: (-11)² - 4×7×(-13) = 121 - (-364) = 121 + 364 = 485 Step 5: Simplify: x = [11 ± √485] / 14 Step 6: Final solutions: x = (11 + √485)/14 and x = (11 - √485)/14

6. Mason is designing a rectangular solar panel. The length of the panel is 9 centimeters longer than its width. The area of the panel is 112 square centimeters. What are the dimensions (width and length) of the solar panel? Answer: width = 7 cm, length = 16 cm Solution: Let w = width in cm. Then length = w + 9. Area = width × length = w(w + 9) = 112.
   Full step-by-step solution

   Step 1: Let w = width in cm. Then length = w + 9. Step 2: Area = width × length = w(w + 9) = 112. Step 3: Expand: w² + 9w = 112. Step 4: Write in standard form: w² + 9w - 112 = 0. Step 5: Identify a = 1, b = 9, c = -112. Step 6: Apply quadratic formula: w = [-b ± sqrt(b² - 4ac)] / (2a). Step 7: Substitute: w = [-9 ± sqrt(9² - 4(1)(-112))] / (2(1)). Step 8: Simplify: w = [-9 ± sqrt(81 + 448)] / 2 = [-9 ± sqrt(529)] / 2. Step 9: sqrt(529) = 23. Step 10: w = [-9 + 23] / 2 = 14/2 = 7, or w = [-9 - 23] / 2 = -32/2 = -16. Step 11: Width cannot be negative, so w = 7 cm. Then length = 7 + 9 = 16 cm. The dimensions are width = 7 cm and length = 16 cm.

7. A company's profit from selling x units of a product is modeled by the quadratic function P(x) = -2x² + 120x - 1000. The company breaks even when profit is zero. How many units must be sold for the company to break even? Answer: A. 50 units Solution: Set the profit function equal to zero for break-even: -2x² + 120x - 1000 = 0 Multiply both sides by -1 to simplify: 2x² - 120x + 1000 = 0 Divide all terms by 2: x² - 60x + 500 = 0 Use the quadratic formula: x = [60 ± sqrt(3600 - 2000)] / 2 Calculate the discriminant: 3600 - 2000 = 1600 Take…
   Full step-by-step solution

   Step 1: Set the profit function equal to zero for break-even: -2x² + 120x - 1000 = 0 Step 2: Multiply both sides by -1 to simplify: 2x² - 120x + 1000 = 0 Step 3: Divide all terms by 2: x² - 60x + 500 = 0 Step 4: Use the quadratic formula: x = [60 ± sqrt(3600 - 2000)] / 2 Step 5: Calculate the discriminant: 3600 - 2000 = 1600 Step 6: Take square root: sqrt(1600) = 40 Step 7: Apply the formula: x = (60 ± 40) / 2 Step 8: Calculate both solutions: x = (60 + 40)/2 = 100/2 = 50, and x = (60 - 40)/2 = 20/2 = 10 Step 9: Both 10 and 50 units give break-even, but the company would choose to sell more units to maximize profit The correct answer is 50 units.

8. Solve: 4x² + 8x - 12 = 0 using quadratic formula Answer: x = 1, x = -3 Solution: Identify coefficients: a = 4, b = 8, c = -12 Apply quadratic formula: x = [-b ± √(b² - 4ac)] / (2a) Substitute values: x = [-8 ± √(8² - 4×4×(-12))] / (2×4) Calculate discriminant: 8² - 4×4×(-12) = 64 - (-192) = 64 + 192 = 256 Calculate square root: √256 = 16 Apply formula: x = [-8 ± 16] / 8…
   Full step-by-step solution

   Step 1: Identify coefficients: a = 4, b = 8, c = -12 Step 2: Apply quadratic formula: x = [-b ± √(b² - 4ac)] / (2a) Step 3: Substitute values: x = [-8 ± √(8² - 4×4×(-12))] / (2×4) Step 4: Calculate discriminant: 8² - 4×4×(-12) = 64 - (-192) = 64 + 192 = 256 Step 5: Calculate square root: √256 = 16 Step 6: Apply formula: x = [-8 ± 16] / 8 Step 7: First solution: x = [-8 + 16]/8 = 8/8 = 1 Step 8: Second solution: x = [-8 - 16]/8 = -24/8 = -3 The solutions are x = 1 and x = -3.