Quadratic Formula

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Quadratic Formula · Grade 9 · Worksheet 3

1. Sophia is designing a parabolic fountain for a park. The height of the water (in meters) above the fountain's nozzle is modeled by the equation h(t) = -4t² + 16t + 6, where t is the time in seconds after the water is ejected. At what time does the water hit the ground (height = 0)? Round your answer to one decimal place. Answer: 4.3 Solution: Set the height equation to zero: -4t² + 16t + 6 = 0. Multiply both sides by -1 to make the leading coefficient positive: 4t² - 16t - 6 = 0.
   Full step-by-step solution

   Step 1: Set the height equation to zero: -4t² + 16t + 6 = 0. Multiply both sides by -1 to make the leading coefficient positive: 4t² - 16t - 6 = 0. Step 2: Identify a = 4, b = -16, c = -6 for the quadratic formula t = [-b ± sqrt(b² - 4ac)] / (2a). Step 3: Calculate the discriminant: b² - 4ac = (-16)² - 4(4)(-6) = 256 + 96 = 352. Step 4: Apply the quadratic formula: t = [16 ± sqrt(352)] / (2 * 4) = [16 ± sqrt(352)] / 8. Step 5: Simplify sqrt(352) = sqrt(16 * 22) = 4 sqrt(22). So t = [16 ± 4 sqrt(22)] / 8 = [4 ± sqrt(22)] / 2. Step 6: Calculate the two possible solutions: t = (4 + sqrt(22)) / 2 and t = (4 - sqrt(22)) / 2. sqrt(22) ≈ 4.6904. So t = (4 + 4.6904) / 2 ≈ 8.6904 / 2 ≈ 4.3452, and t = (4 - 4.6904) / 2 ≈ (-0.6904) / 2 ≈ -0.3452. Step 7: Time cannot be negative, so the water hits the ground at t ≈ 4.3452 seconds. Rounded to one decimal place: t ≈ 4.3 seconds. The answer is 4.3.

2. A company's profit P (in thousands of dollars) from selling x units of a product is modeled by the quadratic equation P = -2x² + 120x - 1000. How many units must the company sell to break even (make zero profit)? Answer: 10 Solution: Set the profit equation equal to zero to find break-even point: -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 discriminant: (-60)² - 4(1)(500) = 3600 -…
   Full step-by-step solution

   Step 1: Set the profit equation equal to zero to find break-even point: -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 discriminant: (-60)² - 4(1)(500) = 3600 - 2000 = 1600 Step 5: Calculate square root: sqrt(1600) = 40 Step 6: Apply 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: The company breaks even at 10 units (the lower quantity where profit first reaches zero) and again at 50 units (where profit returns to zero after peaking). The answer is 10 units.

3. A rectangular garden has a length that is 5 meters more than its width. If the area of the garden is 84 square meters, what are the dimensions of the garden? Answer: width = 7 meters, length = 12 meters Solution: Let the width of the garden be \( w \) meters. The length is 5 meters more than the width, so length \( l = w + 5 \). Area of a rectangle = length × width Given area = 84 square meters.
   Full step-by-step solution

   Let's go step-by-step. --- **Step 1: Define variables** Let the width of the garden be \( w \) meters. The length is 5 meters more than the width, so length \( l = w + 5 \). --- **Step 2: Write the area equation** Area of a rectangle = length × width Given area = 84 square meters. So: \( (w + 5) \times w = 84 \) --- **Step 3: Expand and rearrange** \( w^2 + 5w = 84 \) \( w^2 + 5w - 84 = 0 \) --- **Step 4: Solve the quadratic equation** We solve \( w^2 + 5w - 84 = 0 \) by factoring. Look for two numbers whose product is -84 and whose sum is 5. Possible pairs: (12, -7) → 12 + (-7) = 5, 12 × (-7) = -84. So: \( w^2 + 12w - 7w - 84 = 0 \) \( w(w + 12) - 7(w + 12) = 0 \) \( (w - 7)(w + 12) = 0 \) --- **Step 5: Find possible values of w** \( w - 7 = 0 \) → \( w = 7 \) \( w + 12 = 0 \) → \( w = -12 \) (not valid, width can't be negative) So width \( w = 7 \) meters. --- **Step 6: Find length** Length \( l = w + 5 = 7 + 5 = 12 \) meters. --- **Step 7: Check** Area = length × width = 12 × 7 = 84 square meters. Length (12) is indeed 5 more than width (7). --- **Final answer:** width = 7 meters, length = 12 meters

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 12 units. Write the quadratic equation that models this situation using the Pythagorean theorem, then solve it to find the possible values for x. Answer: 4√5 Solution: The Pythagorean theorem states that in any right triangle, the square of the hypotenuse equals the sum of the squares of the other two sides. This relationship creates quadratic equations when one side length is unknown.
   Full step-by-step solution

   The Pythagorean theorem states that in any right triangle, the square of the hypotenuse equals the sum of the squares of the other two sides. This relationship creates quadratic equations when one side length is unknown. Solving these equations often involves simplifying square roots.

5. 7x² - 11x + 4 = 0 Answer: x = 1, x = 4/7 Solution: Identify coefficients: a = 7, b = -11, c = 4 Calculate discriminant: b² - 4ac = (-11)² - 4(7)(4) = 121 - 112 = 9 Apply quadratic formula: x = [11 ± √9] / (2×7) = [11 ± 3] / 14 Calculate both solutions: x = (11 + 3)/14 = 14/14 = 1, x = (11 - 3)/14 = 8/14 = 4/7 Final solutions: x = 1 and x = 4/7
   Full step-by-step solution

   Step 1: Identify coefficients: a = 7, b = -11, c = 4 Step 2: Calculate discriminant: b² - 4ac = (-11)² - 4(7)(4) = 121 - 112 = 9 Step 3: Apply quadratic formula: x = [11 ± √9] / (2×7) = [11 ± 3] / 14 Step 4: Calculate both solutions: x = (11 + 3)/14 = 14/14 = 1, x = (11 - 3)/14 = 8/14 = 4/7 Step 5: Final solutions: x = 1 and x = 4/7

6. A rectangular garden has a length that is 3 meters longer than its width. If the area of the garden is 40 square meters, what are the dimensions of the garden? Answer: 5 meters by 8 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 this 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 = 40 square meters: \[ w \times (w + 3) = 40 \] --- **Step 3: Expand and rearrange** \[ w^2 + 3w = 40 \] Subtract 40 from both sides: \[ w^2 + 3w - 40 = 0 \] --- **Step 4: Solve the quadratic equation** We can factor it: We need two numbers whose product is -40 and whose sum is 3. Those numbers are 8 and -5. So: \[ (w + 8)(w - 5) = 0 \] --- **Step 5: Find possible values of w** \[ w + 8 = 0 \quad \text{or} \quad w - 5 = 0 \] \[ w = -8 \quad \text{or} \quad w = 5 \] --- **Step 6: Interpret the solutions** Width cannot be negative, so \( w = 5 \) meters. Then length \( l = w + 3 = 5 + 3 = 8 \) meters. --- **Step 7: Check** Area = \( 5 \times 8 = 40 \) square meters. Length (8 m) is indeed 3 m longer than width (5 m). --- **Final answer:** Width = 5 meters, Length = 8 meters.

7. 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.

8. Mere is designing a rectangular fountain for a city park. The length of the fountain is 6 meters more than twice its width. The area of the fountain is 140 square meters. What are the dimensions (width and length) of the fountain? Answer: Width = 8 meters, Length = 22 meters Solution: Let w represent the width in meters. The length is 6 meters more than twice the width, so length = 2w + 6. Area = length × width = (2w + 6) × w = 140.
   Full step-by-step solution

   Step 1: Let w represent the width in meters. The length is 6 meters more than twice the width, so length = 2w + 6. Step 2: Area = length × width = (2w + 6) × w = 140. Step 3: Expand: 2w² + 6w = 140. Step 4: Rewrite in standard form: 2w² + 6w - 140 = 0. Step 5: Divide the entire equation by 2 to simplify: w² + 3w - 70 = 0. Step 6: Use the quadratic formula: w = [-b ± sqrt(b² - 4ac)] / (2a), where a = 1, b = 3, c = -70. Step 7: Calculate the discriminant: b² - 4ac = 3² - 4(1)(-70) = 9 + 280 = 289. Step 8: sqrt(289) = 17. Step 9: Substitute into the formula: w = [-3 ± 17] / (2 × 1) = [-3 ± 17] / 2. Step 10: Find the two solutions: w = (-3 + 17)/2 = 14/2 = 7; w = (-3 - 17)/2 = -20/2 = -10. Step 11: Discard the negative solution because width cannot be negative. So width = 7 meters. Step 12: Length = 2w + 6 = 2(7) + 6 = 14 + 6 = 20 meters. Step 13: Check: Area = 7 × 20 = 140 square meters. Correct. The dimensions of the fountain are width = 7 meters and length = 20 meters.

9. 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