Most students check their work by reading it again. That almost never works: you re-read what you meant to write, not what you actually wrote, and a re-read can't catch an error in your method at all.
Real checking means testing the answer independently of how you got it. Here's how to do that for the problem types you'll actually meet.
1. Substitute your answer back in
The most underused check in all of algebra. If you solved an equation, put the answer back into the original equation — not into a line halfway through your working, which may already contain the mistake.
This turns a solving problem into an arithmetic problem, and arithmetic is much easier to get right. It works for linear equations, quadratics, simultaneous equations, and most word problems.
2. Differentiate to check an integral
Integration and differentiation are inverses, and differentiation is by far the easier direction. So every integral comes with a free check:
If differentiating your answer doesn't return the original integrand, your integral is wrong — no exceptions, no ambiguity. This is the single most reliable check in calculus, and it works no matter how ugly the integral was.
Check the antiderivative by differentiating first, then evaluate the limits. That separates a calculus mistake from an arithmetic one, so you know which part to fix.
3. Test an identity with numbers
When you've simplified or expanded an expression, pick a number, put it into both the original and your final version, and compare. If they're genuinely equivalent, every input gives the same output.
Avoid 0 and 1 as test values — they hide too many errors, because multiplying and adding by them is unusually forgiving. Use something like 2, 3, or 5.
One caveat: passing this test for a single value strongly suggests you're right but doesn't prove it. Two different values passing makes it near-certain for the kind of algebra you meet in homework.
4. Sanity-check the size
Before anything else, ask whether the answer is even plausible. This catches the catastrophic errors — a misplaced decimal, a dropped minus sign, a unit conversion in the wrong direction.
- A probability must sit between 0 and 1. Not 1.4, not −0.2.
- A length, area, mass, or time can't be negative.
- If a car travels for 2 hours at roughly 60 km/h, the answer should be near 120 km — not 12, not 1200.
- In physics, check the units carry through. If you wanted a speed and your units came out as metres, the setup was wrong.
Rough estimation before you solve is even better: decide roughly what the answer should be, then check your exact result lands near it.
5. Hunt for extraneous solutions
Some operations — squaring both sides, multiplying by a variable, taking logs — can invent solutions that don't satisfy the original equation. The algebra is valid; the extra root is still wrong.
Square both sides: x + 6 = x², so x² − x − 6 = 0, giving (x − 3)(x + 2) = 0 and roots x = 3 and x = −2. Now test both in the original:
- x = 3: √9 = 3 ✓ — valid.
- x = −2: √4 = 2, but the right-hand side is −2. Since 2 ≠ −2, this root is extraneous and must be discarded.
The answer is x = 3 alone. Squaring introduced the false root, because squaring destroys sign information.
Any time you square both sides, cancel a variable factor, or take a logarithm, testing your roots in the original equation isn't optional — it's part of the method.
6. Use sum and product of roots
For a quadratic ax² + bx + c = 0, the two roots always satisfy:
That gives you a check that's completely independent of how you solved it. Take 2x² + 3x − 2 = 0, whose roots are x = 1/2 and x = −2:
- Sum: 1/2 + (−2) = −3/2, and −b/a = −3/2 ✓
- Product: (1/2)(−2) = −1, and c/a = −2/2 = −1 ✓
Both match, so the roots are right. If only one matched, you'd know you had one root correct and one wrong — which is far more useful than simply knowing something's off.
Building the habit
Checking feels like it costs time you don't have during a test. In practice it buys time back, because the alternative is losing marks on work you'd already done correctly except for one slip.
A workable rule: check every answer that took more than two minutes to produce, and every answer you felt unsure about. Those are where mistakes concentrate.
It's also worth knowing that this is exactly why Aheadia runs its solutions through real math engines rather than trusting a language model's first attempt — the same principle as substituting back, applied automatically. An answer you can verify is worth more than an answer you simply received.
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