Quadratic Formula

lessonbunny.com

Answer Key & Explanations

Quadratic Formula · Grade 9 · Worksheet 3

1. 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

2. 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

3. Solve: 5x² - 13x + 6 = 0 using quadratic formula Answer: x = 2, x = 3/5 Solution: Identify coefficients: a = 5, b = -13, c = 6 Calculate discriminant: b² - 4ac = (-13)² - 4(5)(6) = 169 - 120 = 49 Apply quadratic formula: x = [13 ± √49] / (2×5) Simplify square root: √49 = 7 x = (13 + 7)/10 = 20/10 = 2 x = (13 - 7)/10 = 6/10 = 3/5 Final solutions: x = 2, x = 3/5
   Full step-by-step solution

   Step 1: Identify coefficients: a = 5, b = -13, c = 6 Step 2: Calculate discriminant: b² - 4ac = (-13)² - 4(5)(6) = 169 - 120 = 49 Step 3: Apply quadratic formula: x = [13 ± √49] / (2×5) Step 4: Simplify square root: √49 = 7 Step 5: Calculate both solutions: x = (13 + 7)/10 = 20/10 = 2 x = (13 - 7)/10 = 6/10 = 3/5 Step 6: Final solutions: x = 2, x = 3/5

4. Solve: 7x² + 12x - 27 = 0 using quadratic formula Answer: x = 1.2857, x = -3 Solution: Identify coefficients: a = 7, b = 12, c = -27 Calculate discriminant: b² - 4ac = 12² - 4(7)(-27) = 144 + 756 = 900 Apply quadratic formula: x = [-12 ± √900] / (2×7) Simplify: x = [-12 ± 30] / 14 Calculate first solution: x = [-12 + 30]/14 = 18/14 = 9/7 ≈ 1.2857 Calculate second solution: x =…
   Full step-by-step solution

   Step 1: Identify coefficients: a = 7, b = 12, c = -27 Step 2: Calculate discriminant: b² - 4ac = 12² - 4(7)(-27) = 144 + 756 = 900 Step 3: Apply quadratic formula: x = [-12 ± √900] / (2×7) Step 4: Simplify: x = [-12 ± 30] / 14 Step 5: Calculate first solution: x = [-12 + 30]/14 = 18/14 = 9/7 ≈ 1.2857 Step 6: Calculate second solution: x = [-12 - 30]/14 = -42/14 = -3 Final answer: x = 9/7 (approximately 1.2857) and x = -3

5. Solve: 2x² + 11x - 21 = 0 using quadratic formula. Answer: x = 3/2, x = -7 Solution: Identify coefficients: a = 2, b = 11, c = -21 Quadratic formula: x = [-b ± sqrt(b² - 4ac)] / (2a) Substitute: x = [-11 ± sqrt(11² - 4(2)(-21))] / (2(2)) Compute discriminant: 11² = 121, 4(2)(-21) = -168, so 121 - (-168) = 121 + 168 = 289 sqrt(289) = 17 x = [-11 ± 17] / 4 First solution: x = (-11…
   Full step-by-step solution

   Step 1: Identify coefficients: a = 2, b = 11, c = -21 Step 2: Quadratic formula: x = [-b ± sqrt(b² - 4ac)] / (2a) Step 3: Substitute: x = [-11 ± sqrt(11² - 4(2)(-21))] / (2(2)) Step 4: Compute discriminant: 11² = 121, 4(2)(-21) = -168, so 121 - (-168) = 121 + 168 = 289 Step 5: sqrt(289) = 17 Step 6: x = [-11 ± 17] / 4 Step 7: First solution: x = (-11 + 17)/4 = 6/4 = 3/2 Step 8: Second solution: x = (-11 - 17)/4 = -28/4 = -7 The solutions are x = 3/2 and x = -7.

6. A company's profit from selling x units of a product is modeled by the quadratic function P(x) = -2x² + 120x - 1600. The company breaks even when profit equals zero. How many units must be sold for the company to break even? Answer: 20 Solution: Set the profit function equal to zero: -2x² + 120x - 1600 = 0 Multiply both sides by -1 to make the leading coefficient positive: 2x² - 120x + 1600 = 0 Divide all terms by 2 to simplify: x² - 60x + 800 = 0 Apply the quadratic formula: x = [60 ± sqrt(60² - 4(1)(800))] / 2(1) Calculate the…
   Full step-by-step solution

   Step 1: Set the profit function equal to zero: -2x² + 120x - 1600 = 0 Step 2: Multiply both sides by -1 to make the leading coefficient positive: 2x² - 120x + 1600 = 0 Step 3: Divide all terms by 2 to simplify: x² - 60x + 800 = 0 Step 4: Apply the quadratic formula: x = [60 ± sqrt(60² - 4(1)(800))] / 2(1) Step 5: Calculate the discriminant: 3600 - 3200 = 400 Step 6: Take the square root: sqrt(400) = 20 Step 7: Substitute into the formula: x = [60 ± 20] / 2 Step 8: Calculate both solutions: x = (60 + 20)/2 = 80/2 = 40, and x = (60 - 20)/2 = 40/2 = 20 Step 9: Since the company breaks even when profit is zero, both 20 and 40 units would give zero profit. However, the company would typically want the lower number to break even, so the answer is 20 units.

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. A physics class is testing projectile motion by launching a ball from a catapult. The ball's height above ground (in meters) is modeled by the equation h(t) = -5t² + 20t + 1.5, where t is time in seconds. At what time does the ball hit the ground? Answer: 4.1 Solution: Set up the equation for when the ball hits the ground: -5t² + 20t + 1.5 = 0 Use the quadratic formula: t = [-b ± sqrt(b² - 4ac)] / (2a), where a = -5, b = 20, c = 1.5 Calculate the discriminant: b² - 4ac = 20² - 4(-5)(1.5) = 400 + 30 = 430 Apply the quadratic formula: t = [-20 ± sqrt(430)] / (2…
   Full step-by-step solution

   Step 1: Set up the equation for when the ball hits the ground: -5t² + 20t + 1.5 = 0 Step 2: Use the quadratic formula: t = [-b ± sqrt(b² - 4ac)] / (2a), where a = -5, b = 20, c = 1.5 Step 3: Calculate the discriminant: b² - 4ac = 20² - 4(-5)(1.5) = 400 + 30 = 430 Step 4: Apply the quadratic formula: t = [-20 ± sqrt(430)] / (2 × -5) = [-20 ± sqrt(430)] / -10 Step 5: Calculate the two possible solutions: t = [-20 + sqrt(430)] / -10 ≈ [-20 + 20.736] / -10 ≈ 0.736 / -10 ≈ -0.0736 (not valid, time cannot be negative) and t = [-20 - sqrt(430)] / -10 ≈ [-20 - 20.736] / -10 ≈ -40.736 / -10 ≈ 4.0736 Step 6: Round to one decimal place: t ≈ 4.1 seconds The ball hits the ground at approximately 4.1 seconds.

9. 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