How to check quadratic equation answers
Use this page when you already have a quadratic answer and want to verify it properly before moving on.
Start here if you want the short version before reading the full method.
- The quickest reliable check is to substitute each root back into the original equation and see whether the result is 0.
- If you factorised, expand the brackets again. If you used the quadratic formula, compare the result with the discriminant as a second sanity check.
What this topic means and what to look for first.
Checking a quadratic answer is not just about catching arithmetic slips. It also tells you whether the method and the structure of the equation agree with each other.
A strong check is usually shorter than the original solve, so it is worth treating as part of the process, not as an optional extra.
One reliable route through the topic.
- 1Take one solution at a time and substitute it into the original equation, not a rearranged version.
- 2Simplify carefully and confirm the result is 0.
- 3If the quadratic came from brackets, expand the brackets to make sure they rebuild the original expression exactly.
- 4Use the discriminant or graph interpretation as a final reasonableness check when helpful.
See the method in action.
Check x = -2 for x^2 + 5x + 6 = 0
- Substitute -2 into the equation: (-2)^2 + 5(-2) + 6.
- This becomes 4 - 10 + 6.
- The result is 0, so x = -2 is a valid root.
Check whether (2x + 1)(x + 3) matches 2x^2 + 7x + 3
- Expand the brackets: 2x^2 + 6x + x + 3.
- Combine like terms to get 2x^2 + 7x + 3.
- Because the expansion matches exactly, the factorisation is correct.
Things that commonly send the method off track.
- Checking the root in a rearranged line instead of in the original equation.
- Substituting a negative value without brackets and changing the sign structure accidentally.
- Assuming one correct root means the second root must also be correct.
Use a short verification pass before moving on.
- Always check both roots separately if the quadratic has two solutions.
- Use brackets around negative substitutions so the signs remain correct.
- If the discriminant said there should be two real roots, make sure your checked result also reflects that.
Try a few variations before switching to a calculator or solver tool.
- Check x = 3 for x^2 - 6x + 9 = 0
- Check x = 1/2 for 2x^2 + 3x - 2 = 0
- Expand (x - 4)(x + 3) and compare it with x^2 - x - 12
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Short answers worth checking.
Substituting the root back into the original equation is usually the fastest reliable check.
Yes. Each root should be checked separately because one correct line earlier in the method does not guarantee both final roots are correct.
Yes. Expanding the brackets is often enough to confirm that the factorisation is correct.
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