Digital SATMath: Advanced MathHard
A robotics engineer is programming a robot arm. The path of the arm's gripper can be described by the equation y = x^2 - 6x + 8, and a laser sensor detects objects along the line y = 2x - 7. At what x-coordinate(s) will the robot arm's gripper intersect the laser sensor's path?
- Ax = 4
- Bx = 3
- Cx = 4 and x = 5
- Dx = 3 and x = 5
Show answer & explanationAnswer & explanation
Correct answer: D. x = 3 and x = 5
To find the intersection points, set the two equations equal to each other: x^2 - 6x + 8 = 2x - 7. Rearrange into a standard quadratic equation: x^2 - 6x - 2x + 8 + 7 = 0, which simplifies to x^2 - 8x + 15 = 0. Factor the quadratic equation: (x - 3)(x - 5) = 0. This yields two possible x-coordinates: x - 3 = 0 => x = 3, and x - 5 = 0 => x = 5. These are the x-coordinates where the paths intersect.
Why the other options are wrong
- A. Incorrect x-value; likely a factoring error or incorrect solution to the quadratic.
- B. Only one of the correct x-values; incomplete answer.
- C. One correct x-value and one incorrect x-value.
Solving Systems of Nonlinear Equations (Quadratic & Linear)
To find the intersection points of a quadratic function and a linear function, set their y-expressions equal to each other, forming a quadratic equation. Solve this quadratic equation for x, which may yield zero, one, or two real solutions.
- Set y-values equal: y1 = y2.
- Rearrange into standard quadratic form: ax^2 + bx + c = 0.
- Solve for x using factoring, quadratic formula, or completing the square.
- Each x-solution corresponds to an intersection point.
Memory trick: 'Equal-ize' the paths to find where they 'meet'!