Quadratic Formula Worksheets Grade 9

1. The quadratic function f(x) = 2x² - 8x + 3 can be written in vertex form as f(x) = a(x - h)² + k. What is the value of k?
2. A local farmer is building a rectangular chicken coop against the side of her barn, so she only needs to use fencing for three sides. She has 40 meters of fencing material available. If the side perpendicular to the barn has a length of 'x' meters, write a quadratic function in standard form, A(x), that represents the area of the chicken coop in terms of x. Then, determine the maximum possible area she can enclose.
3. Aroha is helping her uncle, a structural engineer, design a parabolic arch for a new greenhouse. The shape of the arch can be modeled by the general quadratic equation ax² + bx + c = 0, where the solutions represent the x-intercepts (the points where the arch meets the ground). To find these critical points, Aroha needs to derive the quadratic formula from the standard form. Show all the steps to derive the quadratic formula x = [-b ± √(b²-4ac)] / 2a by completing the square on the general quadratic equation ax² + bx + c = 0.

…and 4 more problems

1. A rectangular community garden is being designed with a length that is 7 meters more than its width. The total area of the garden needs to be 170 square meters to accommodate all the planned vegetable beds. What is the width of the garden in meters?
2. A right triangle is drawn on a coordinate plane with vertices at (0,0), (x,0), and (0,8). The hypotenuse has length 17 units. Using the Pythagorean theorem, derive the quadratic equation that models this situation in standard form ax² + bx + c = 0.
3. Derive x = [-b ± √(b²-4ac)] / (2a) by completing the square on ax² + bx + c = 0

…and 4 more problems

1. Mere is helping her class derive the quadratic formula from the general quadratic equation ax² + bx + c = 0, where a ≠ 0. By completing the square, show all steps to derive the formula x = [-b ± √(b² - 4ac)] / (2a).
2. x² + 5x + 6 = 0
3. Isabella is a structural engineer working on a bridge design. The main support cable of the bridge forms a parabolic shape that can be modeled by the general quadratic equation ax² + bx + c = 0, where a, b, and c are constants that depend on the bridge's dimensions and load. Isabella needs to derive a universal formula for finding the x-intercepts (points where the cable crosses the support beam) of any such parabolic cable, expressed solely in terms of a, b, and c. By completing the square on the general quadratic equation ax² + bx + c = 0, derive the quadratic formula that Isabella can use.

…and 5 more problems