9514 1404 393
Answer:
C and D
Step-by-step explanation:
Extraneous solutions arise in a couple of cases, both of which are illustrated in this list of answer choices.
1. Square roots (and other even-index radicals). The square root is defined only for positive arguments. When the radical is removed by squaring the equation, the possibility of negative arguments is introduced. Hence, there may be extraneous solutions.
<u>Example</u>:
x = √(x +2) ⇒ x² = x+2 . . . has 2 roots, one of which is extraneous
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2. Rational function equations. Such an equation is only defined when the denominator is not zero. A method often taught for solution of these equations is "cross-multiplying"--essentially multiplying by the product of the denominators. In the case where this product is not the least common denominator, the equation will have an extra factor that can introduce an extraneous root.
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The answer choices C and D are likely to have extraneous roots based on the above observations.
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<em>Additional comment</em>
Choice D would usually be solved by cross-multiplying. That would give the quadratic equation (x -3)² = 0, which only has solutions x=3. The denominators are zero at x=3, so these solutions are extraneous. If the right side is subtracted from the left side, the fractions can be combined to give ...
((x+3) -6)/((x -3)(x +3)) = 0 ⇒ 1/(x+3) = 0 . . . . no solution
