Quadratic Formula Worksheets Grade 9

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.
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?
3. Derive x = [-b ± √(b²-4ac)] / 2a by completing square on ax²+bx+c=0

…and 2 more problems

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).
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?
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.

…and 1 more problems

1. Liam is deriving the quadratic formula by completing the square on the general quadratic equation ax² + bx + c = 0, where a ≠ 0. He visualizes the process as a geometric area model: he starts with a large square of side length x representing the x² term, then adds a rectangle of area bx (where b = 10) along one side, and finally includes a constant area c = 25. He wants to complete the square by adding a small square to form a perfect square trinomial. Show the step-by-step derivation of the quadratic formula x = [-b ± √(b² - 4ac)] / (2a) from ax² + bx + c = 0 using the method of completing the square, using the specific numbers a = 5, b = 10, c = 25 as a concrete example to illustrate each step, then generalize.
2. x² + 5x + 6 = 0
3. Tane is a mathematician working on a new model for projectile motion. He starts with the general quadratic equation in standard form: ax² + bx + c = 0, where a, b, and c are real numbers and a ≠ 0. Tane wants to derive the quadratic formula, x = [-b ± sqrt(b² - 4ac)] / (2a), by completing the square on this general equation. Show all the steps Tane must follow to complete this derivation.

…and 3 more problems