How to Check Your Math Homework and Answers

Why checking is a skill worth practising

Many students treat checking as rereading their working, which rarely finds anything: the eye sees what it expects to see. Real checking uses a different route to the same answer, so that an error on the first route shows up as a disagreement. The methods below each provide such a route. Most take less than a minute, and together they catch the large majority of homework and test errors.

Checking also builds understanding. Substituting an answer back into an equation reminds you what a solution actually is. Differentiating an integral reminds you how the two operations are linked. Over time, students who check habitually develop a sense for when an answer looks wrong, which is one of the most valuable instincts in mathematics.

1. Substitute the answer back

A solution to an equation is a value that makes it true, so the most direct check is to plug it in. Use the original equation, not a later line of your working, because an error in an early step would otherwise go unnoticed.

Substitution is essential when you squared both sides or multiplied by an expression containing the variable, because those steps can create extraneous solutions. Solving $\sqrt{x + 7} = x - 5$ by squaring gives $x^2 - 11x + 18 = 0$, so $x = 9$ or $x = 2$. Substituting shows $\sqrt{16} = 4 = 9 - 5$, which works, but $\sqrt{9} = 3$ while $2 - 5 = -3$, so $x = 2$ must be rejected.

2. Reverse the operation

Many operations have an inverse that undoes them, and the inverse is often easier to perform. Factoring is checked by expanding, division by multiplying, and integration by differentiating.

• Factoring: if you wrote $x^2 + x - 12 = (x + 4)(x - 3)$, expand the right side: $x^2 - 3x + 4x - 12 = x^2 + x - 12$.
• Division: if $391 \div 17 = 23$, check that $17 \times 23 = 391$.
• Integration: if $\int 2x\cos(x^2)\,dx = \sin(x^2) + C$, differentiating $\sin(x^2)$ with the chain rule gives $2x\cos(x^2)$, as required.
• Logarithms: if $\log_2 32 = 5$, check that $2^5 = 32$.
• Inverse functions: if $f^{-1}$ is correct, then $f(f^{-1}(x)) = x$ for several test values.

3. Estimate before and after

A rough estimate made before calculating gives a target. Rounding $49 \times 21$ to $50 \times 20 = 1000$ tells you the exact answer, 1029, is plausible, while 10,290 or 129 would be clear mistakes. Estimation is especially useful with calculators, where a mistyped digit or misplaced decimal point produces an answer of completely the wrong size.

In applied problems, ask whether the answer makes sense in context. A person cannot be 25 metres tall, a probability cannot be negative, and a car cannot average 900 km per hour. These reality checks take seconds and catch some of the most expensive errors.

4. Use the structure of the answer

Many results have relationships you can check without redoing the problem. For a quadratic $ax^2 + bx + c = 0$ with roots $r_1$ and $r_2$, the sum and product of the roots must satisfy

For $x^2 - 5x + 6 = 0$, the roots 2 and 3 have sum 5 and product 6, matching $-\frac{-5}{1}$ and $\frac{6}{1}$. Other structural checks include symmetry, signs and degree: the product of a degree 2 and a degree 3 polynomial must have degree 5, a determinant of a matrix with two equal rows must be zero, and a probability distribution must add to 1.

5. Test special or simple values

When your answer is an expression or identity rather than a number, test it at convenient values. If you simplified an expression, the original and simplified versions must agree at every allowed input. Try $x = 0$, $x = 1$ and $x = 2$; a mismatch at any of them proves an error.

Derivatives can be checked numerically with a small step. For $f(x) = x^3$ the derivative is $3x^2$, so $f'(2) = 12$. The difference quotient $\frac{2.001^3 - 1.999^3}{0.002} = 12.000001$ agrees closely, confirming the result. The same approach works for limits and for integrals estimated with a few rectangles.

6. Check units and dimensions

In science and applied math, units are a free check. Area must come out in square units, volume in cubic units, and speed in distance per time. If a formula for an area gives an answer in metres rather than square metres, a factor is missing. Writing units through every line of a calculation makes such errors visible immediately.

7. Solve it another way

For important problems, a second method is the strongest check. A system of equations solved by substitution can be re-solved by elimination; a quadratic solved by factoring can be checked with the quadratic formula; an integral can be compared with a numerical estimate. If two independent methods agree, the answer is very likely right.

Graphing provides a visual alternative. The solutions of an equation are where the two sides' graphs intersect, and the roots of a polynomial are its x-intercepts. A sketch, or the graphing calculator, shows at a glance whether you have the right number of solutions in roughly the right places.

8. Compare with a step-by-step solver

After finishing, compare your work with a step-by-step tool such as the equation solver, the factoring calculator or the derivative calculator. If the answers differ, do not simply copy the tool's answer. Find the first line where the steps diverge; that line contains your error, and understanding it is what prevents it next time. For a catalogue of the most common errors, see how to avoid careless math mistakes.

Use the cheapest check first. Substitution and estimation take seconds; re-solving takes minutes. Save the expensive checks for high-value problems or when a quick check fails.

A checking routine for timed tests

Under exam conditions there is rarely time to re-solve everything, so checking has to be selective. As you work, mark any question where you felt unsure, made a long calculation or squared both sides. When you reach the end, return to the marked questions first. For each one, do a single fast check, usually substitution or estimation, and move on unless it fails. Spend any time that remains rereading the wording of every question to make sure you answered exactly what was asked, in the requested form, units and rounding. This routine catches the most errors per minute of checking time.

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