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Worksheet: Solving Radical Equations with a Variable on Each Side

Q1:

Consider the following argument for solving βˆ’ 3 = √ π‘₯ :

  1. βˆ’ 3 = √ π‘₯
  2. ( βˆ’ 3 ) = ο€Ί √ π‘₯  2 2
  3. 9 = π‘₯ .

What mistake was made?

  • Anot checking for extraneous solutions
  • Bsquaring βˆ’ 3 does not give 9

Q2:

In order to solve the equation π‘₯ + √ π‘₯ = 6 , Michael started with

What solutions did Michael find?

Q3:

The natural way to solve √ π‘₯ + 9 βˆ’ 2 π‘₯ = 3 introduces an extraneous solution. What is that extraneous solution?

  • A π‘₯ = βˆ’ 3 4
  • B π‘₯ = βˆ’ 2
  • C π‘₯ = 1 1 7
  • D π‘₯ = βˆ’ 1 1 4
  • E π‘₯ = 3

Q4:

For what value of π‘˜ is π‘₯ = 1 introduced as an extraneous solution to the equation π‘₯ = √ π‘₯ + 3 + π‘˜ when it is solved in the natural way?