Digital SATMath: AlgebraMedium
A robotics engineer is programming a robot's movement. The robot's position, P(t), in meters, at time t seconds, is described by the radical equation P(t) = 5 + √(2t + 1). At what time 't' will the robot be at a position of 8 meters?
- A4 seconds
- B6 seconds
- C2 seconds
- D8 seconds
Show answer & explanationAnswer & explanation
Correct answer: A. 4 seconds
Set P(t) = 8: 8 = 5 + √(2t + 1). Subtract 5 from both sides: 3 = √(2t + 1). Square both sides to eliminate the radical: 3^2 = (√(2t + 1))^2, which gives 9 = 2t + 1. Subtract 1 from both sides: 8 = 2t. Divide by 2: t = 4 seconds. Always check for extraneous solutions: P(4) = 5 + √(2*4 + 1) = 5 + √(8 + 1) = 5 + √9 = 5 + 3 = 8. The solution is valid.
Why the other options are wrong
- B. This value might be a result of miscalculation, such as 2t = 12 instead of 8.
- C. This value might result from errors in squaring or solving the linear equation.
- D. This value might result from adding 5 to 8 instead of subtracting, or other algebraic errors.
Solving Radical Equations
To solve an equation containing a radical expression, isolate the radical, then raise both sides to the power corresponding to the index of the radical (e.g., square for square root).
- Isolate the radical term on one side of the equation.
- Raise both sides to the power of the radical's index.
- Solve the resulting linear or quadratic equation.
- Always check for extraneous solutions by substituting back into the original equation.
Memory trick: Radical Road: Isolate the root, square it, then solve, but always check!