Study guide
7.2 Solving Radical Equations and Extraneous Roots
ACCUPLACER Advanced Algebra and Functions Exam
Track your progress as you read
Finished sections get a check mark.
1 Introduction & Exam Overview
1.1 Current Exam Facts 1.2 Scoring, Skills Insight Bands, and Placement 1.3 How to Study for AAF
2 Linear Equations
2.1 Writing Linear Equations in One and Two Variables 2.2 Solving Linear Equations 2.3 Linear Inequalities 2.4 Systems of Two Linear Equations
3 Linear Applications and Graphs
3.1 Slope, Intercepts, and Graphing Lines 3.2 Parallel and Perpendicular Lines 3.3 Modeling with Linear Functions 3.4 Graphing Linear Inequalities and Systems
4 Factoring
4.1 GCF, Grouping, and Special Products 4.2 Factoring Quadratic Trinomials 4.3 Factoring Cubics and Higher Polynomials
5 Quadratics
5.1 Quadratic Forms, Graphs, Vertex, and Intercepts 5.2 Solving Quadratics by Factoring 5.3 Quadratic Formula and Completing the Square 5.4 Quadratic Inequalities and Quadratic-Linear Systems
6 Functions
6.1 Function Notation, Evaluation, Domain, and Range 6.2 Graphs of Functions, Vertical Line Test, Maxima and Minima 6.3 Function Operations, Composition, and Inverses 6.4 Transformations and Interpreting Functions in Context
7 Radical and Rational Equations
7.1 Simplifying Radical Expressions 7.2 Solving Radical Equations and Extraneous Roots 7.3 Rational Expressions and Equations 7.4 Graphs, Domain, and Range of Radical and Rational Functions
8 Polynomial Equations
8.1 Polynomial Operations and Structure 8.2 Solving Polynomial Equations 8.3 Graphing Polynomials, Roots, and End Behavior
9 Exponential and Logarithmic Equations
9.1 Exponential Functions, Growth, and Decay 9.2 Graphs of Exponential Functions 9.3 Logarithms and Logarithmic Properties 9.4 Solving Exponential and Logarithmic Equations
10 Geometry Concepts for Algebra 1
10.1 Area, Perimeter, and Prism Volume 10.2 Distance Formula and Pythagorean Theorem 10.3 Dilations, Rotations, Translations, and Reflections
11 Geometry Concepts for Algebra 2
11.1 Volume of Cylinders, Cones, and Spheres 11.2 Intersecting Lines and Angle Relationships 11.3 Triangle Similarity and Congruence 11.4 Circles in the Coordinate Plane
12 Trigonometry
12.1 Right-Triangle Trigonometry and Special Triangles 12.2 Unit Circle, Radians, and Arc Length 12.3 Trigonometric Graphs and Identities 12.4 Solving Trig Equations; Law of Sines and Cosines
7.2 Solving Radical Equations and Extraneous Roots
Key Takeaways Explain This Topic AI
Isolate the radical, then square; always substitute each candidate into the original equation because squaring can create extraneous roots.
For even roots the radicand must be ≥ 0; a principal square root is never negative, so an isolated equation √(3x+1) = −5 has no real solution.
√(2x+3)+1 = 6 isolates to √(2x+3) = 5, squares to x = 11, and 11 checks in the original.
√(x+6) = x yields quadratic candidates x = 3 and x = −2; only x = 3 satisfies the original.
Two-radical √(2x+9) = √x + 3 has solutions x = 0 and x = 36 after isolating, squaring, and checking both.
Last updated: August 2026
7.2 Solving Radical Equations and Extraneous Roots
Simplifying a radical is not the same as solving a radical equation. Table 11’s Radical and rational equations slice includes creating and solving those equations. On AAF, a typical item isolates a square root, squares both sides, and then — the step that separates the 237–249 radical-equations band from guessing — checks every candidate in the original equation . Squaring is not reversible. It can manufacture extraneous roots .
Skills Insight band 237–249 includes solving simple radical equations. Later draws mix in two radicals, a radical with a linear term, or a nested root, and the same domain discipline still decides which candidates survive. Work FREE items at /practice/accuplacer-advanced-algebra . Domain language from Function Notation, Evaluation, Domain, and Range is assumed: for an even root, the radicand must be ≥ 0 . Simplifying after you solve still uses Simplifying Radical Expressions .
Domain before algebra
For √(g(x)) to be a real number, g(x) ≥ 0 . For ⁿ√(g(x)) with even n , the same restriction holds. Cube roots have no such restriction: ∛(x − 1) is defined for every real x .
Write the domain before you square. Candidates that solve the squared equation but miss the domain are discarded. Candidates that sit in the domain still have to satisfy the original equation, because squaring can turn a false statement into a true one.
- Equation Domain restriction
- √(2x + 3) + 1 = 6 2x + 3 ≥ 0 , so x ≥ −3/2
- √(3x + 1) + 7 = 2 3x + 1 ≥ 0 , and after isolation the right side is negative
- √(x + 6) = x x ≥ −6 , and because a principal square root is ≥ 0, also x ≥ 0
- ∛(2x − 1) = 3 all real x
The principal (AAF) square root never returns a negative . If, after isolating, you have √(something) = a negative number, stop. There is no real solution . Squaring anyway wastes time and produces an extraneous number you will have to throw away.
Isolate, then square
A reliable sequence:
State the domain of every even-root radicand.
Isolate one radical on one side (add or subtract first; then divide if a coefficient sits outside the radical).
Square both sides (or cube, if the isolated root is a cube root).
Solve the resulting linear or quadratic equation.
Check every candidate in the original equation. Keep only the values that work.
If a coefficient is outside, isolate that too: 2√(x + 1) = 10 becomes √(x + 1) = 5 before squaring. Squaring too early on 2√(x + 1) = 10 happens to survive because both sides are nonnegative, but 2√(x + 1) + 3 = 7 squared too early expands into a mess that hides the simple √(x + 1) = 2 .
Worked example: √(2x + 3) + 1 = 6
Isolate: √(2x + 3) = 5 .
Domain: 2x + 3 ≥ 0 ⇒ x ≥ −3/2 . The isolated right side 5 is positive, so a real solution is possible.
Square: 2x + 3 = 25 .
2x = 22 .
x = 11 .
Check in the original: √(2·11 + 3) + 1 = √25 + 1 = 5 + 1 = 6 . True. And 11 ≥ −3/2 . Solution: x = 11 .
A nearby trap is forgetting to isolate and squaring √(2x + 3) + 1 as if it were √(2x + 3) + 1^2 , or squaring termwise into 2x + 3 + 1 = 36 . Another trap is √(2x + 3) + 1 = 0 , which isolates to √(2x + 3) = −1 . No real solution — do not square.
The isolation trap: a square root never equals a negative
Consider √(3x + 1) + 7 = 2 .
Isolate: √(3x + 1) = 2 − 7 = −5 .
The left side is a principal square root, so it is ≥ 0. It cannot equal −5. No real solution.
If you square blindly: 3x + 1 = 25 , x = 8 . Plug back: √(24 + 1) + 7 = 5 + 7 = 12 , which is not 2. The candidate x = 8 is extraneous . The faster path is to notice the isolated radical equaled a negative and stop.
This is the same structural trap as an item that isolates to a negative after moving a large constant, but the numbers here are original. Train the reflex: look at the sign after isolation before you square .
Extraneous roots from squaring
Squaring is not a one-to-one operation. a = b implies a^2 = b^2 , but a^2 = b^2 also holds when a = −b . That is why a candidate can satisfy the squared equation and fail the original.
Worked example: √(x + 6) = x
Domain: x + 6 ≥ 0 ⇒ x ≥ −6 . Also the right side must be ≥ 0 because it equals a principal square root, so x ≥ 0 .
Square: x + 6 = x^2 .
x^2 − x − 6 = 0 .
(x − 3)(x + 2) = 0 .
Candidates: x = 3 and x = −2 .
Check x = 3 : √9 = 3 . True. Keep.
Check x = −2 : √4 = −2 would require 2 = −2 . False. Discard. (It also fails x ≥ 0 .)
Only x = 3. Reporting both factors of the quadratic is the most common error on this item type. Factoring is still correct algebra for the squared equation; the original equation is stricter.
Two radicals
When two square roots appear, isolate one of them, square, then isolate the remaining radical and square again. Check in the original at the end — twice squaring can add even more extras.
Worked example: √(2x + 9) = √x + 3
Domain: 2x + 9 ≥ 0 and x ≥ 0 , so x ≥ 0 . The right side √x + 3 is automatically ≥ 3, hence nonnegative, so isolation is already legal.
Square: 2x + 9 = (√x + 3)^2 = x + 6√x + 9 .
Simplify: 2x + 9 = x + 6√x + 9 .
x = 6√x .
At this stage either x = 0 or, moving everything left, x − 6√x = 0 . Let u = √x (so u ≥ 0 and x = u^2 ): u^2 − 6u = 0 ⇒ u(u − 6) = 0 ⇒ u = 0 or u = 6 . Then x = 0 or x = 36 .
Check x = 0 : √9 = 0 + 3 ⇒ 3 = 3 . Keep.
Check x = 36 : √(72 + 9) = 6 + 3 ⇒ √81 = 9 ⇒ 9 = 9 . Keep.
Both x = 0 and x = 36 work. Two-radical items are not automatically “one extra, one keeper.” Checking decides. A related trap expands (√x + 3)^2 as x + 9 , dropping the middle term 6√x , and then never recovers a linear-in- √x equation.
Nested radicals
A nested (composition) equation still follows isolate-then-raise-to-a-power, once per layer.
Worked example: √(5 + √(x − 2)) = 3
Domain: inner x − 2 ≥ 0 ⇒ x ≥ 2 , and outer 5 + √(x − 2) ≥ 0 (always true once the inner root exists).
Square once: 5 + √(x − 2) = 9 .
√(x − 2) = 4 .
Square again: x − 2 = 16 .
x = 18 .
Check: √(5 + √16) = √(5 + 4) = √9 = 3 . True. x = 18 , and 18 ≥ 2 .
If the first isolation had produced √(5 + √(x − 2)) = −3 , you would stop with no real solution, just as in the monomial case. Nested does not change the sign rule.
Creating a radical equation
Table 11 also wants creating the equation from a sentence. “The square root of three more than twice a number, increased by 1, is 6” is exactly √(2x + 3) + 1 = 6 . “A side length is the square root of an area of 50” is s = √50 , which you would simplify to 5√2 using the previous section. Creating and solving are sequential skills: write the equation with the radicand grouped in parentheses, then isolate.
Cube roots
Cube both sides, not square. ∛(2x − 1) = 3 ⇒ 2x − 1 = 27 ⇒ x = 14 . Check: ∛27 = 3 . Cube roots can equal negatives: ∛(2x − 1) = −2 ⇒ 2x − 1 = −8 ⇒ x = −7/2 , which is valid. The “never negative” rule is for even roots only. Cubing is one-to-one on the reals, so cube-root equations do not manufacture the same style of extraneous roots that squaring does — you still check, because of arithmetic slips, but the domain is all reals.
What the CAT is really testing
The 237–249 band’s simple radical equation is usually one isolated square root and one linear result, possibly with no real solution after isolation. Higher bands add the second radical or a quadratic after squaring. Either way, the check is not optional . The diagram below is the whole method: domain, isolate, refuse a negative even root, raise to a power, solve, and sort keepers from extras in the original equation. Graphing those keepers as points on y = √(g(x)) is Graphs, Domain, and Range of Radical and Rational Functions .