Quadratic Formula

lessonbunny.com

Answer Key & Explanations

Quadratic Formula · Grade 9 · Worksheet 1

1. Hana is an aerospace engineer designing a parabolic satellite dish. The cross-section of the dish is modeled by the general quadratic equation y = ax² + bx + c, where y is the vertical distance from the base of the dish (in meters) and x is the horizontal distance from the center (in meters). To find the points where the dish meets the ground (y = 0), Hana needs a formula that works for any set of coefficients a, b, and c. Starting from the standard quadratic equation ax² + bx + c = 0, where a ≠ 0, derive the quadratic formula by completing the square. Show all algebraic steps clearly. Answer: x = [-b ± sqrt(b² - 4ac)] / (2a) Solution: Start with the general quadratic equation in standard form: ax² + bx + c = 0, where a ≠ 0. Subtract c from both sides to isolate the terms with x: ax² + bx = -c.
   Full step-by-step solution

   Step 1: Start with the general quadratic equation in standard form: ax² + bx + c = 0, where a ≠ 0. Step 2: Subtract c from both sides to isolate the terms with x: ax² + bx = -c. Step 3: Divide both sides by a to make the coefficient of x² equal to 1: x² + (b/a)x = -c/a. Step 4: Complete the square on the left side. Take half of the coefficient of x, which is (b/a) / 2 = b/(2a). Square it: (b/(2a))² = b²/(4a²). Add this term to both sides: x² + (b/a)x + b²/(4a²) = -c/a + b²/(4a²). Step 5: 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 6: Combine the terms on the right side. Write -c/a with denominator 4a²: -c/a = -4ac/(4a²). So the right side becomes: (b² - 4ac)/(4a²). Thus: (x + b/(2a))² = (b² - 4ac)/(4a²). Step 7: Take the square root of both sides. Remember to include both positive and negative roots: x + b/(2a) = ± sqrt((b² - 4ac)/(4a²)). Step 8: Simplify the square root on the right: sqrt((b² - 4ac)/(4a²)) = sqrt(b² - 4ac) / (2|a|). Since we have a ± sign and a can be positive or negative, we write: x + b/(2a) = ± sqrt(b² - 4ac) / (2a). Step 9: Solve for x by subtracting b/(2a) from both sides: x = -b/(2a) ± sqrt(b² - 4ac) / (2a). Step 10: Combine the terms over the common denominator 2a: x = [-b ± sqrt(b² - 4ac)] / (2a). The final derived formula is x = [-b ± sqrt(b² - 4ac)] / (2a).

2. A community center is designing a rectangular meditation garden with a stone path of uniform width around it. The garden itself measures 8 meters by 6 meters. The total area including the path is 120 square meters. What is the width of the stone path in meters? Answer: 2 Solution: Let w represent the width of the stone path in meters. The garden is 8m by 6m. With the path, the total length becomes 8 + 2w and the total width becomes 6 + 2w.
   Full step-by-step solution

   Step 1: Let w represent the width of the stone path in meters. Step 2: The garden is 8m by 6m. With the path, the total length becomes 8 + 2w and the total width becomes 6 + 2w. Step 3: The total area including the path is given as 120 square meters, so we write the equation: (8 + 2w)(6 + 2w) = 120. Step 4: Expand the left side: 48 + 16w + 12w + 4w² = 120. Step 5: Simplify: 4w² + 28w + 48 = 120. Step 6: Subtract 120 from both sides: 4w² + 28w - 72 = 0. Step 7: Divide the entire equation by 4 to simplify: w² + 7w - 18 = 0. Step 8: Factor the quadratic: (w + 9)(w - 2) = 0. Step 9: Solve for w: w = -9 or w = 2. Step 10: Since width cannot be negative, the solution is w = 2. The width of the stone path is 2 meters.

3. Derive x = [-b ± √(b²-4ac)] / 2a by completing square on ax²+bx+c=0 Answer: x = [-b ± √(b²-4ac)] / 2a Solution: Start with the general quadratic equation: ax² + bx + c = 0 Divide through by a: x² + (b/a)x + c/a = 0 Move constant term to right side: x² + (b/a)x = -c/a Complete the square: Take half of (b/a), square it to get (b/2a)² = b²/4a² Add this to both sides: x² + (b/a)x + b²/4a² = -c/a + b²/4a²…
   Full step-by-step solution

   Step 1: Start with the general quadratic equation: ax² + bx + c = 0 Step 2: Divide through by a: x² + (b/a)x + c/a = 0 Step 3: Move constant term to right side: x² + (b/a)x = -c/a Step 4: Complete the square: Take half of (b/a), square it to get (b/2a)² = b²/4a² Step 5: Add this to both sides: x² + (b/a)x + b²/4a² = -c/a + b²/4a² Step 6: Write left side as perfect square: (x + b/2a)² = -c/a + b²/4a² Step 7: Combine right side over common denominator: (x + b/2a)² = (-4ac + b²)/4a² Step 8: Take square root of both sides: x + b/2a = ±√(b² - 4ac)/2a Step 9: Solve for x: x = [-b ± √(b² - 4ac)] / 2a

4. Matiu is designing a parabolic mirror for a solar concentrator. The shape of the mirror is described by the quadratic equation y = ax² + bx + c, where y is the vertical distance from the vertex. To find the precise focal point, Matiu needs to solve for x when y = 0. Derive the quadratic formula x = [-b ± √(b² - 4ac)] / (2a) by completing the square on the general quadratic equation ax² + bx + c = 0, where a, b, and c are constants and a ≠ 0. Answer: x = [-b ± sqrt(b² - 4ac)] / (2a) Solution: Start with the general quadratic equation: ax² + bx + c = 0. Subtract c from both sides: ax² + bx = -c. Divide both sides 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 both sides by a (since a ≠ 0): x² + (b/a)x = -c/a. Step 4: To complete the square, add (b/(2a))² to both sides: x² + (b/a)x + (b/(2a))² = -c/a + (b/(2a))². Step 5: Simplify the left side as a perfect square: (x + b/(2a))² = -c/a + b²/(4a²). Step 6: Combine the terms on 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) = ± sqrt(b² - 4ac) / (2a). Step 8: Solve for x by subtracting b/(2a) from both sides: x = -b/(2a) ± sqrt(b² - 4ac) / (2a). Step 9: Combine the terms over the common denominator 2a: x = [-b ± sqrt(b² - 4ac)] / (2a). Thus, the quadratic formula is x = [-b ± sqrt(b² - 4ac)] / (2a).

5. A drone is launched from a platform 12 meters high. Its height above ground is modeled by the function h(t) = -5t² + 20t + 12, where t is time in seconds. The drone needs to drop a package when it reaches its maximum height. How many seconds after launch should the package be dropped? Answer: 2 Solution: We are given the height function: h(t) = -5t² + 20t + 12. This is a quadratic function in the form h(t) = at² + bt + c, with a = -5, b = 20, c = 12.
   Full step-by-step solution

   We are given the height function: h(t) = -5t² + 20t + 12. This is a quadratic function in the form h(t) = at² + bt + c, with a = -5, b = 20, c = 12. Since a < 0, the parabola opens downward, so the maximum height occurs at the vertex. For a quadratic function at² + bt + c, the t-coordinate of the vertex is: t = -b / (2a) Substitute b = 20 and a = -5: t = -20 / (2 * -5) t = -20 / (-10) t = 20 / 10 t = 2 So the drone reaches maximum height at t = 2 seconds after launch. Therefore, the package should be dropped 2 seconds after launch.