Card 1 of 20
EasyStudy card 1
Prompt
What is the first step in most radical equations?
Answer
Isolate one radical expression.
MathChapter 13: Polynomial and Radical Functions
Flashcards
Solving Radical Equations Flashcards
Twenty recall cards on isolation, power operations, domains, candidate checks, and extraneous solutions.
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Card 1 of 20. Answer hidden.
Card 2 of 20
EasyStudy card 2
Prompt
How is an isolated square root removed?
Answer
Square both complete sides.
Card 3 of 20
EasyStudy card 3
Prompt
How is an isolated cube root removed?
Answer
Cube both complete sides.
Card 4 of 20
EasyStudy card 4
Prompt
Why are solutions to a squared equation called candidates?
Answer
Squaring can create values that do not satisfy the original equation.
Card 5 of 20
EasyStudy card 5
Prompt
What is an extraneous solution?
Answer
A value produced during algebra that fails the original equation.
Card 6 of 20
EasyStudy card 6
Prompt
Where must every radical-equation candidate be checked?
Answer
In the original, unsquared equation.
Card 7 of 20
EasyStudy card 7
Prompt
What sign must a principal square root have?
Answer
It is always nonnegative.
Card 8 of 20
MediumStudy card 8
Prompt
What domain condition applies to \(\sqrt{g(x)}\)?
Answer
\(g(x)\ge0\).
Card 9 of 20
MediumStudy card 9
Prompt
Should only the radical term be squared?
Answer
No. Square each entire side of the equation.
Card 10 of 20
MediumStudy card 10
Prompt
What is \((a+b)^2\)?
Answer
\(a^2+2ab+b^2\).
Card 11 of 20
MediumStudy card 11
Prompt
Why is the cross term important after squaring?
Answer
Omitting \(2ab\) changes the equation and its candidates.
Card 12 of 20
MediumStudy card 12
Prompt
Solve \(\sqrt{x+5}=4\).
Answer
\(x=11\), and substitution confirms it.
Card 13 of 20
MediumStudy card 13
Prompt
If \(\sqrt{x+1}=x-1\), what sign condition must the right side satisfy?
Answer
\(x-1\ge0\), so \(x\ge1\).
Card 14 of 20
MediumStudy card 14
Prompt
Does a negative candidate always fail a radical equation?
Answer
No. It fails only if it violates the domain or original equality.
Card 15 of 20
MediumStudy card 15
Prompt
Can cube-root equations create sign-based extraneous roots merely from cubing?
Answer
Cubing is one-to-one over the reals, though algebra and domain checks remain prudent.
Card 16 of 20
MediumStudy card 16
Prompt
What if an equation contains two radicals?
Answer
Isolate one, apply a power, simplify, then isolate the remaining radical if needed.
Card 17 of 20
HardStudy card 17
Prompt
Why can \(\sqrt{x+6}=x\) reject a negative algebraic candidate?
Answer
The left side is nonnegative, so a negative right side cannot match it.
Card 18 of 20
HardStudy card 18
Prompt
What is the safest four-step workflow?
Answer
Isolate, apply the matching power, solve, then check every candidate in the original equation.
Card 19 of 20
HardStudy card 19
Prompt
How can domain analysis save time?
Answer
It can eliminate impossible candidates before lengthy substitution, but final checks still verify equality.
Card 20 of 20
HardStudy card 20
Prompt
What is the key logical risk when squaring both sides?
Answer
The implication can gain solutions because opposite-signed expressions have equal squares.
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