Discriminant Analysis Worksheets Grade 9

1. Emma is analyzing the trajectory of a drone she is testing. The height of the drone above the ground, in meters, is modeled by the quadratic function h(t) = -3t² + 21t + 27, where t is time in seconds. Emma wants to know if the drone will ever reach a height of exactly 75 meters during its flight. Using the discriminant of the quadratic formula, determine whether this height is achievable.
2. Sophia is designing a parabolic arch for a new garden entrance. The shape of the arch can be modeled by the equation h(x) = -2x² + 28x, where h(x) represents the height in feet at a horizontal distance x feet from the left base. She wants to know if the arch will be tall enough to allow a 100-foot-tall sculpture to pass through the center. Using the discriminant of the quadratic equation, determine whether the arch ever reaches a height of exactly 100 feet.
3. Mason launches a model rocket from the ground with an initial upward velocity of 30 m/s. The height of the rocket above the ground, in meters, is modeled by the quadratic function h(t) = -5t² + 30t, where t is the time in seconds after launch. Isabella, his classmate, claims the rocket will reach a height of exactly 50 meters during its flight. Using the discriminant of the quadratic equation, determine whether Isabella is correct.

…and 3 more problems

1. For the quadratic equation 14x² - 35x + 21 = 0, calculate the discriminant and determine how many real solutions exist.
2. Mere sketches the graph of the quadratic function y = 3x² + 12x + k on a coordinate plane. The parabola opens upward and its vertex lies exactly on the x-axis. Determine the value of k and state how many real x-intercepts the graph has.
3. A company's profit from selling x units of a product is modeled by the quadratic function P(x) = -2x² + 120x - 1000. The company needs to determine how many units they must sell to break even (profit equals zero). Using the quadratic formula, find the number of units that represents the break-even point.

…and 5 more problems

1. Tane is designing a parabolic skateboard ramp for a community park. The height of the ramp (in meters) above the ground at a horizontal distance x meters from the start is modeled by the quadratic function h(x) = -2x² + 24x + 18. The park committee requires that the ramp must reach a height of exactly 90 meters at some point to qualify for a national competition. Using the discriminant of the quadratic formula, determine whether this height is achievable.
2. Kaia is analyzing the graph of a quadratic function. The parabola opens downward and its equation is y = -3x² + 12x - 12. The vertex of the parabola lies directly on the x-axis. Calculate the discriminant of this quadratic equation and state how many real x-intercepts the graph has.
3. Noah is testing a new water balloon launcher. The height of a water balloon, in meters, after t seconds is modeled by the quadratic function h(t) = -5t² + 28t + 8. Noah wants to know if the water balloon will ever reach a height of exactly 50 meters during its flight. Using the discriminant of the quadratic equation, determine whether this height is achievable.

…and 4 more problems