Here is the habit. You simplify something radical — say \sqrt{x^2 + 6x + 9} — and claim the answer is x + 3. Before you circle it and move on, stop. Pick x = 2. Plug it into the original: \sqrt{4 + 12 + 9} = \sqrt{25} = 5. Plug it into your simplified form: 2 + 3 = 5. They match. Go. If they hadn't matched, you would stop right there and find the error, because the match is the only thing standing between you and a silent wrong answer on the exam paper.
This is a close cousin of the general "plug in x = 1 or 2" sanity check, tuned specifically for radicals, where the error rate is much higher because squaring and square-rooting interact badly with signs, absolute values, and fractional exponents.
Why x = 2 or x = 3 specifically
The right value exercises every part of the expression without being a root of any sub-expression.
- x = 0 is too trivial. Most terms vanish; almost nothing is tested.
- x = 1 is too clean. Every power of x equals 1, so x, x^2, and x^{3/2} are indistinguishable. Coefficient mistakes hide here.
- x = 2 produces distinct powers — 2, 4, 8, 16 — so each term contributes a different recognisable number.
- x = 3 is the backup when x = 2 happens to be a root of something in the expression. It gives 3, 9, 27, 81, again all distinct.
Avoid any value that zeros out a factor. If your expression contains (x - 3), picking x = 3 makes that factor zero — both sides are trivially zero, and the check is vacuous. Scan first, pick second.
Worked example 1 — \sqrt{x^2 + 6x + 9}
The claim is that this simplifies to |x + 3|. The absolute value is there because the square root of a square is always non-negative, and x + 3 can be negative.
Check at x = 2. Original: \sqrt{4 + 12 + 9} = \sqrt{25} = 5. Simplified: |2 + 3| = 5. Match.
Check at x = -5. This is the interesting one — it is the regime where the absolute value matters. Original: \sqrt{25 - 30 + 9} = \sqrt{4} = 2. Simplified: |{-5} + 3| = |-2| = 2. Match.
If you had mistakenly written the simplified form as x + 3 (without the absolute value), the check at x = -5 would have given -2 for the simplified form against 2 for the original — mismatch, caught. The plug-in check is what makes you remember the absolute value even when your instinct is to drop it.
Worked example 2 — a wrong simplification
A student simplifies \sqrt{18x^2} and writes 18x. Let's run the check.
At x = 2. Original: \sqrt{18 \cdot 4} = \sqrt{72} \approx 8.49. Claimed simplified form: 18 \cdot 2 = 36. Huge mismatch — 36 is more than four times the right answer.
The size of the mismatch is a hint. A factor-of-four miss on a radical usually means you forgot a square root somewhere. Scanning back, the student sees the error: they treated \sqrt{18x^2} as 18x instead of taking the square root of 18x^2 as a whole. The correct simplification is \sqrt{18} \cdot \sqrt{x^2} = 3\sqrt{2} \cdot |x|, which equals 3x\sqrt{2} for x \geq 0.
Re-check at x = 2. Corrected form: 3 \cdot 2 \cdot \sqrt{2} = 6\sqrt{2} \approx 8.49. Match. Answer recovered, total cost of the detour is perhaps forty seconds.
Worked example 3 — a messier case you get right
Expand (\sqrt{x} + 1)^2 + (\sqrt{x} - 1)^2. You compute:
The \sqrt{x} terms cancel — clean. But the cancellation is exactly the kind of step where a sign error silently creeps in. Check it.
At x = 4. Original: (\sqrt{4} + 1)^2 + (\sqrt{4} - 1)^2 = 3^2 + 1^2 = 9 + 1 = 10. Simplified: 2 \cdot 4 + 2 = 10. Match.
At x = 9. Original: (3 + 1)^2 + (3 - 1)^2 = 16 + 4 = 20. Simplified: 2 \cdot 9 + 2 = 20. Match.
Two independent matches. High confidence. Commit.
Worked example 4 — catching a cancellation mistake
Simplify \dfrac{\sqrt{x^2}}{x} for x \neq 0. A tempting move: \sqrt{x^2} = x, so the whole thing is x / x = 1. Let's check.
At x = 2. Original: \sqrt{4} / 2 = 2/2 = 1. Simplified: 1. Match.
At x = -2. Original: \sqrt{4} / (-2) = 2 / (-2) = -1. Simplified: 1. Mismatch. Caught.
The mistake was assuming \sqrt{x^2} = x. Actually \sqrt{x^2} = |x|. So the correct simplification is |x|/x, which is +1 when x > 0 and -1 when x < 0 — what mathematicians call \mathrm{sign}(x). A single positive-value check would have missed this; the negative-value re-check exposed it. When your expression involves squaring or rooting, test both signs.
Choosing "safe" values
A short rulebook.
- Default to x = 2.
- If the expression contains (x - 2), switch to x = 3.
- If it contains x^2 - 4 = (x-2)(x+2), avoid both x = \pm 2; pick x = 3.
- If there is a \sqrt{x}, pick x = 4 or x = 9 for clean arithmetic (avoid 0 and 1 — too weak).
- If you suspect a sign error (anything involving \sqrt{x^2}, absolute values, or even-index roots), run the check at a positive value and a negative value.
When one check is enough, and when to use two
One check catches most real mistakes, because real mistakes — a dropped sign, a missed absolute value, a forgotten coefficient — move the value at most inputs. For second-degree polynomial identities, two matching values constitute a proof. For higher-degree or radical expressions, two matches aren't a proof but are overwhelming evidence. When in doubt, run a second check at a different value. Thirty extra seconds, near-certainty.
What this habit cannot do
An honest limitation list.
- It is not a proof. Two different expressions can coincidentally agree at one point — x^2 and 2x both equal 4 at x = 2, yet they're not the same. Single-point agreement is strong evidence, not a certificate.
- It does not judge which form is "simpler". The check verifies equality at a test point, nothing more. Picking x\sqrt{18} versus 3x\sqrt{2} as a final answer is a separate question about convention.
- It may miss domain issues. A simplification valid for x \geq 0 can fail for x < 0; a positive-only test misses it. Test negatives too whenever the expression squares, roots, or takes absolute values.
When you can skip the check
Not every simplification needs a check. For 3x + 5x = 8x, it's overkill. Reserve the habit for cases where the error rate is high:
- Expressions involving squaring or square-rooting — especially the \sqrt{x^2} = |x| trap.
- Expressions with absolute values or sign-sensitive cancellations.
- Expressions with fractional exponents, where the law-of-exponents sanity check applies.
- Any multi-step chain where an early sign error would silently propagate.
The 30-second investment
Plugging in a value takes about thirty seconds and catches the large majority of real mistakes, because real mistakes move the numerical value of the expression. The alternative — trusting your algebra without a check — is a false economy. The way a senior developer writes a unit test right after writing the function, not because they distrust themselves but because the cost of checking is tiny and the cost of not checking is everything.
Close
Your simplifications are claims: "this expression equals that expression, for every legal input". You don't verify "for every input" — that's infinite. You verify at one carefully chosen point. If the two forms agree there, you have strong evidence; if they disagree, the claim is definitely wrong. Cheap insurance. Make it a reflex every time a radical sign, an absolute value, or a fractional exponent shows up in your working.