You have finished the problem. You have an answer written down. You have no idea whether it is right, and the back of the book only prints the odd numbers.
Most students respond by working the problem again, which is the weakest thing you can do. If you made a mistake the first time, you will usually make the same one the second time, because you will follow the same reasoning down the same path. What you need is a check that comes at the answer from a different direction. There are five, they take about two minutes between them, and they catch different kinds of error.
Substitute it back into the original problem
This is the only check that tests your answer against the question instead of against your own work, which makes it the one to do first.
So you have finished the problem and you are sitting on a pair of roots. Do not admire them. Feed them straight back in and watch what happens. This is the one check in the whole list that interrogates the question rather than interrogating you, which is exactly why it goes first.
Take 2x² − 5x − 3 = 0, where the roots came out as x = 3 and x = −1/2.
For x = 3: 2(9) − 15 − 3 = 18 − 18 = 0.
For x = −1/2: 2(1/4) + 5/2 − 3 = 0.5 + 2.5 − 3 = 0.
Both land on zero, so both are really roots. You know now, instead of hoping.
One word matters more than the rest of this section: original. Substitute into the problem as it was handed to you, not into the version you rewrote in line four. If you introduced an error while rearranging, checking against your rearranged line will confirm the error rather than catch it.
Here is where that stops being pedantry and starts costing marks. Take the equation √(x + 6) = x and work it the obvious way:
- Square both sides: x + 6 = x²
- Rearrange: x² − x − 6 = 0
- Factor: (x − 3)(x + 2) = 0
- Read off the answers: x = 3 and x = −2
Both of those genuinely solve the squared equation. Only one solves the equation you were handed. Try x = 3 and √9 is 3, which is fine. Try x = −2 and √4 is 2, and 2 is not −2. That second root is extraneous. Nothing in your working would ever have caught it, because your working was correct. Only substitution catches it.
Run the operation backwards
Every operation you know has an inverse, and the inverse is a check.
Divided something? Multiply back. Took a square root? Square it. Factored a polynomial? Expand the factors and see whether you get what you started with, which takes about ten seconds and catches sign errors reliably.
Calculus is the cleanest case. If you compute an integral and get sin(x²) + C, differentiate it: you get cos(x²) · 2x. If that matches the integrand you started with, you are done, and no answer key was involved. Integration is genuinely hard to do and easy to verify, which is an unusual and very useful asymmetry.
Differentiation runs the same way in reverse, though the check is often more work than the original. If the uncertainty is not really about your arithmetic but about which rule the expression called for, that is a different diagnosis and needs a different fix, and I have written about how to tell which rule a function is asking for separately.
Estimate first, then compare
Before you calculate anything, guess roughly what the answer should be. Write the guess down. It costs five seconds and it catches the errors that are most embarrassing to hand in.
Take 47 × 19. Round both numbers and you are looking at 50 × 20, which is 1000. So the answer lives somewhere near a thousand.
Work it out properly and you get 893, which is where it should be. But suppose you had come out with 89, or with 8,930. The estimate flags either one immediately. Re-reading your working would not have, because a misplaced decimal looks perfectly reasonable sitting there on the page.
Estimation cannot tell you an answer is right. It reliably tells you an answer is wrong, and it is the only check that works when you are three lines into a problem and have not finished yet.
Check the units, then check the question
Two failures live here, and they are different.
The first is units. If the question asks how long the trip takes and your answer is 45 miles, you have solved for the wrong quantity, and no amount of correct arithmetic saves it. Track units through the calculation and they will tell you what you have produced.
The second failure is subtler and costs more marks. You solved something correctly, but not the thing that was asked. The question said "how many more tickets did the second showing sell" and you wrote down the total for the second showing. Every step was right. The answer is still wrong.
Word problems are where all of this piles up, because translation is the step where the question quietly stops being the question. Reread the final sentence before you write anything down. Out loud, if nobody is around. Does what you have match what it actually asked for? If turning a written situation into equations is the bit that keeps going sideways, the trouble is translation, not algebra, and it wants practising on its own.
Solve it a second way
George Pólya built this into the final stage of his method. The looking-back step asks three questions. Can you check the result? Can you check the argument? Can you get there another way? That third one is the strongest check you have. A second method fails differently from the first, so when two of them land in the same place, the agreement is telling you something real.
Factored a quadratic? Push it through the formula as well.
For 2x² − 5x − 3 = 0 the formula gives x = [5 ± √(25 + 24)]/4, which is [5 ± 7]/4, which is 3 and −1/2.
Those are the same two answers, reached by a route that shares almost nothing with factoring. Factoring depends on spotting a pattern. The formula depends on grinding arithmetic. When two methods that different agree with each other, you have stopped guessing.
I reach for graphs more than anything else here, and most students never think of it. Say your answer to x² = 4x − 3 was x = 1 and x = 3. Put y = x² and y = 4x − 3 on the same axes and see where they cross. If the crossings sit at 1 and 3, your algebra has been confirmed through a channel that involves no algebra at all. And plotting both sides together is faster than reworking the problem from the top, which is the thing that makes people skip checking in the first place.
When the two methods disagree
Sooner or later they will, and the instinct is to run the first method again. Resist it. You already know that method's output, and repeating it mostly just re-confirms whatever you did.
Do this instead. Take both routes and find the earliest line where they stop agreeing. Not the final answers, the intermediate values. One of them will have a step that the other one does not, or a sign that flipped, and the disagreement is always born at a specific line rather than spread across the whole page. Compare from the top down and the fork is usually obvious within thirty seconds.
Then work out which side was wrong and why. "I dropped a negative when I distributed" is a fixable, nameable error you will recognise next week. "I got it wrong somehow" is not, and it will happen again.
Two minutes, on every problem
Nobody runs all five checks on everything. In practice, substitute back and estimate, always. They cost almost nothing and between them they catch the majority of what goes wrong. Add the second-method check on anything that is worth real marks, and on anything where you had a moment of uncertainty partway through, since that feeling is usually correct.
The reason this matters beyond the grade is what it does to your relationship with the subject. A student who can verify their own work does not need to be told whether they are right. That is a different experience of mathematics from the one where you hand in an answer and wait to find out, and it is available to you tonight, on the very next problem you finish.