Digital SATMath: Advanced MathMedium
A robotics engineer is programming a robot arm. The path of the arm's gripper can be described by the function h(x) = -0.5x^2 + 4x - 6, where h(x) is the vertical height and x is the horizontal distance, both in meters. At what horizontal distances does the gripper touch the ground (i.e., h(x) = 0)?
- Ax = 0 and x = 8
- Bx = 1 and x = 5
- Cx = 3 and x = 4
- Dx = 2 and x = 6
Show answer & explanationAnswer & explanation
Correct answer: D. x = 2 and x = 6
To find where the gripper touches the ground, set h(x) = 0: -0.5x^2 + 4x - 6 = 0. Multiply by -2 to simplify: x^2 - 8x + 12 = 0. Factor the quadratic: (x - 2)(x - 6) = 0. This gives solutions x = 2 and x = 6.
Why the other options are wrong
- A. x=0 is from setting the constant to zero, not solving the equation. x=8 is a plausible distractor.
- B. These are plausible distractors, implying incorrect factoring or quadratic formula application.
- C. These are plausible distractors, implying incorrect factoring or quadratic formula application.
Roots of a Quadratic Equation
The roots (or x-intercepts, or zeros) of a quadratic equation ax^2 + bx + c = 0 are the values of x for which the function equals zero. They can be found by factoring, completing the square, or using the quadratic formula.
- Quadratic formula: x = [-b ± √(b^2 - 4ac)] / (2a).
- Factoring is possible if roots are rational.
- The discriminant (b^2 - 4ac) determines the nature of the roots.
Memory trick: Set Function to Zero, Solve for X!