Quadratic Formula

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Answer Key & Explanations

Quadratic Formula · Grade 9 · Worksheet 2

1. A student named Noah is exploring a visual pattern of squares. The first square has side length 1 unit, the second square has side length 2 units, and the third square has side length 3 units. He notices that the area of the nth square is n². He wants to derive the quadratic formula by completing the square on the general quadratic equation ax² + bx + c = 0. Starting with the equation ax² + bx + c = 0, where a ≠ 0, show all steps to derive the quadratic formula x = [-b ± √(b² - 4ac)] / (2a). Answer: x = [-b ± √(b² - 4ac)] / (2a) Solution: Start with the general quadratic equation: ax² + bx + c = 0. Divide every term by a (since a ≠ 0): x² + (b/a)x + (c/a) = 0. Move the constant term to the right side: x² + (b/a)x = -c/a.
   Full step-by-step solution

   Step 1: Start with the general quadratic equation: ax² + bx + c = 0. Step 2: Divide every term by a (since a ≠ 0): x² + (b/a)x + (c/a) = 0. Step 3: Move the constant term to the right side: x² + (b/a)x = -c/a. Step 4: Complete the square. Take half of the coefficient of x, which is (b/a)/2 = b/(2a). Square it: (b/(2a))² = b²/(4a²). Add this to both sides: x² + (b/a)x + b²/(4a²) = -c/a + b²/(4a²). Step 5: The left side is now a perfect square: (x + b/(2a))² = -c/a + b²/(4a²). Step 6: Combine the right side over a common denominator of 4a²: -c/a = -4ac/(4a²). So the right side becomes: (-4ac + b²)/(4a²) = (b² - 4ac)/(4a²). Step 7: Take the square root of both sides: x + b/(2a) = ± √[(b² - 4ac)/(4a²)]. Step 8: Simplify the square root: √[(b² - 4ac)/(4a²)] = √(b² - 4ac) / √(4a²) = √(b² - 4ac) / (2|a|). Since the ± sign already accounts for sign, we write: x + b/(2a) = ± √(b² - 4ac) / (2a). Step 9: Isolate x: x = -b/(2a) ± √(b² - 4ac) / (2a). Step 10: Combine over the common denominator 2a: x = [-b ± √(b² - 4ac)] / (2a). The final derived quadratic formula is x = [-b ± √(b² - 4ac)] / (2a).

2. A rectangular poster for a school science fair has a total area of 96 square inches. The poster's length is 4 inches more than its width. The science club wants to add a border of uniform width around the entire poster to make it more visually appealing. After adding the border, the total area becomes 165 square inches. What is the width of the border in inches? Answer: 1.5 Solution: Let x be the width of the original poster. Then the length is x + 4.
   Full step-by-step solution

   Step 1: Let x be the width of the original poster. Then the length is x + 4. Step 2: The original area is x(x + 4) = 96 Step 3: Solve for x: x² + 4x - 96 = 0 Step 4: Factor: (x + 12)(x - 8) = 0 Step 5: x = 8 (discard negative solution) Step 6: So original width = 8 inches, original length = 12 inches Step 7: Let w be the border width. New dimensions: width = 8 + 2w, length = 12 + 2w Step 8: New area: (8 + 2w)(12 + 2w) = 165 Step 9: Expand: 96 + 16w + 24w + 4w² = 165 Step 10: Simplify: 4w² + 40w + 96 = 165 Step 11: Subtract 165: 4w² + 40w - 69 = 0 Step 12: Use quadratic formula: w = [-40 ± sqrt(40² - 4(4)(-69)] / (2×4) Step 13: w = [-40 ± sqrt(1600 + 1104)] / 8 Step 14: w = [-40 ± sqrt(2704)] / 8 Step 15: w = [-40 ± 52] / 8 Step 16: w = 12/8 = 1.5 or w = -92/8 = -11.5 Step 17: Border width must be positive, so w = 1.5 inches

3. Charlotte is a structural engineer designing a parabolic arch for a new pedestrian bridge. The shape of the arch follows a quadratic curve given by the general equation y = ax² + bx + c, where y is the height in meters and x is the horizontal distance from the left support. To analyze the bridge's load capacity, Charlotte needs to find the x-coordinates where the arch meets the ground (where y = 0). Rather than solving for specific coefficients, she wants to derive a general formula that will work for any parabolic arch of this form. Starting from the standard quadratic equation ax² + bx + c = 0, where a ≠ 0, show step-by-step how to derive the quadratic formula x = [-b ± √(b² - 4ac)] / (2a) using the method of completing the square. Answer: x = [-b ± sqrt(b² - 4ac)] / (2a) Solution: Start with the general quadratic equation: ax² + bx + c = 0, where a ≠ 0. Subtract c from both sides to isolate the x-terms: ax² + bx = -c.
   Full step-by-step solution

   Step 1: Start with the general quadratic equation: ax² + bx + c = 0, where a ≠ 0. Step 2: Subtract c from both sides to isolate the x-terms: ax² + bx = -c. Step 3: Divide both sides by a (since a ≠ 0) to make the coefficient of x² equal to 1: x² + (b/a)x = -c/a. Step 4: Complete the square. Take half of the coefficient of x, which is (b/a)/2 = b/(2a). Square it: (b/(2a))² = b²/(4a²). Step 5: Add this term to both sides of the equation: x² + (b/a)x + b²/(4a²) = -c/a + b²/(4a²). Step 6: The left side is now a perfect square trinomial. Factor it as (x + b/(2a))²: (x + b/(2a))² = -c/a + b²/(4a²). Step 7: Combine the terms on the right side by finding a common denominator (4a²): -c/a = (-4ac)/(4a²). So the right side becomes: (-4ac + b²) / (4a²) = (b² - 4ac) / (4a²). Step 8: Now the equation is: (x + b/(2a))² = (b² - 4ac) / (4a²). Step 9: Take the square root of both sides, remembering the ± sign: x + b/(2a) = ± √[(b² - 4ac) / (4a²)]. Step 10: Simplify the square root on the right: √[(b² - 4ac) / (4a²)] = √(b² - 4ac) / √(4a²) = √(b² - 4ac) / (2|a|). Since squaring eliminates the sign, we can write ± √(b² - 4ac) / (2a). Step 11: Isolate x by subtracting b/(2a) from both sides: x = -b/(2a) ± √(b² - 4ac) / (2a). Step 12: Combine the terms over the common denominator 2a: x = [-b ± √(b² - 4ac)] / (2a). The quadratic formula is derived: x = [-b ± sqrt(b² - 4ac)] / (2a).

4. Emma is deriving the quadratic formula by completing the square on the general quadratic equation ax² + bx + c = 0, where a ≠ 0. She has drawn a large square on a coordinate plane to represent the area model for completing the square. The side length of the square is x, and she adds a rectangle of width b/a to one side to represent the linear term. After dividing the rectangle in half and rearranging, she forms a new square. Following the algebraic steps, what is the final derived formula for x in terms of a, b, and c? Answer: x = [-b ± √(b² - 4ac)] / (2a) Solution: Start with the general quadratic equation: ax² + bx + c = 0. Subtract c from both sides: ax² + bx = -c. Divide every term by a (since a ≠ 0): x² + (b/a)x = -c/a.
   Full step-by-step solution

   Step 1: Start with the general quadratic equation: ax² + bx + c = 0. Step 2: Subtract c from both sides: ax² + bx = -c. Step 3: Divide every term by a (since a ≠ 0): x² + (b/a)x = -c/a. Step 4: Complete the square by adding (b/(2a))² to both sides: x² + (b/a)x + (b/(2a))² = -c/a + (b/(2a))². Step 5: The left side is a perfect square trinomial: (x + b/(2a))² = -c/a + b²/(4a²). Step 6: Combine the right side over a common denominator 4a²: (x + b/(2a))² = (b² - 4ac)/(4a²). Step 7: Take the square root of both sides: x + b/(2a) = ± √(b² - 4ac) / (2a). Step 8: Subtract b/(2a) from both sides: x = -b/(2a) ± √(b² - 4ac) / (2a). Step 9: Combine into a single fraction: x = [-b ± √(b² - 4ac)] / (2a). The quadratic formula is derived as x = [-b ± √(b² - 4ac)] / (2a).