ACT Preparation / Unit 11 ACT 1012 worked example, 60 to 90 seconds
Solving a Radical Equation
The idea
A radical equation is solved in four steps. First, isolate the radical on one side. Second, raise both sides to the power of the index; if a second radical remains, isolate it and repeat. Third, solve the resulting equation. Fourth, substitute every answer into the original equation and discard any that fail.
In practice
Worked example: √(2x + 7) + 4 = x. Step one: √(2x + 7) = x − 4. Step two: 2x + 7 = x² − 8x + 16. Step three: x² − 10x + 9 = 0, so (x − 9)(x − 1) = 0, and x = 9 or x = 1. Step four: x = 9 gives √25 + 4 = 9, true; x = 1 gives √9 + 4 = 7, not 1. Only 9 solves it.
Where people slip
Square the whole side, not term by term: (x − 4)² is x² − 8x + 16, not x² + 16.
Your place on the trail
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