Mistakes · Algebra
5 Common Quadratic Formula Mistakes
Four of the five happen before any real calculating starts, which is why they are so easy to miss on the way back through.
1. Getting the sign of b wrong
The formula opens with −b. When b is already negative, −b is positive, and a great many wrong answers begin right there.
| Equation | b | −b | The trap |
|---|---|---|---|
| x² − 5x + 6 | −5 | +5 | writing −5 instead of +5 |
| x² − x − 1 | −1 | +1 | a bare −x is b = −1 |
| x² + 3x − 4 | +3 | −3 | this one is straightforward |
The check: the two roots must add to −b/a. If b was −5 and a is 1, the roots must sum to 5. Roots summing to −5 mean the sign was dropped.
2. Dividing only part of the numerator by 2a
The entire expression −b ± √D sits over 2a. Writing −b ± √D/2a divides only the root, which is a different and wrong calculation.
- (−4 ± 2√10) ÷ 4correct: the whole numerator over 4
- (−2 ± √10) ÷ 2both terms divided by 2
- −4 ± (2√10 ÷ 4)wrong: only the root was divided
The same error reappears at the simplification stage. If you cancel a factor, it must come out of every term of the numerator.
3. Not putting the equation in standard form
The formula reads its coefficients from ax² + bx + c = 0. Applying it to 3x² = 5x + 2 as though c were 2 gives a confident, wrong answer.
Rearranged properly it is 3x² − 5x − 2 = 0, so b = −5 and c = −2. Both signs changed on the way across.
4. Mishandling 4ac when c is negative
In the discriminant, subtracting a negative adds. For x² − x − 1 the discriminant is 1 − 4(1)(−1) = 1 + 4 = 5, not 1 − 4 = −3.
This one is expensive because it usually turns a solvable equation into an apparently impossible one. If a question with a sensible answer suddenly has no real roots, this is the first place to look.
5. Rounding before the end
√5 is exact. 2.236 is not, and once it enters the working the error travels into both roots and into anything computed from them.
Keep the surd until the final line, and simplify it rather than approximating it. If the question wants a decimal, round once, at the very end, to the precision asked for.
Two more worth naming
Neither of these is quite a sign error, but both cost marks in the same quiet way.
- Missing a repeated root. When the discriminant is zero there is one root, not two. Writing the same value twice is untidy; writing two different values means an arithmetic slip has crept in, because a zero discriminant cannot produce two.
- Forgetting the 2a denominator entirely. When a = 1 the denominator is 2, and it is easy to write nothing at all. That halves both roots, and the answer still looks plausible.
The second is the reason the sum check is worth building in: if the 2a denominator was dropped, the sum of the roots comes out at twice −b/a rather than −b/a, and the error announces itself.
The check that catches all five
Every mistake above is caught by the same two-second test. Add the roots and multiply them: the sum must equal −b/a and the product must equal c/a.
It works because those two relationships come from the coefficients directly rather than from your working, so an arithmetic slip cannot satisfy both. Doing it habitually is faster than re-solving, and it is what separates a careful answer from a lucky one.
One more habit worth the ten seconds: before solving, note whether the discriminant is a perfect square. If it is, the roots should come out rational, and a surd in your final line means something went wrong. If it is not, rational roots are equally suspicious. Knowing what kind of answer to expect is half of checking your answer.
Questions about common quadratic formula mistakes
Why do I keep getting the wrong sign?
Almost always because b was negative and −b was written as negative too. Write the coefficients down separately with their signs before substituting anything.
What if my discriminant comes out negative but the question expects an answer?
Check 4ac first. If c is negative, subtracting 4ac adds, and getting that backwards is the usual cause of a spurious negative discriminant.
Do I need to simplify the surd?
Yes, and then check whether every term of the fraction shares a factor. √48 becomes 4√3, and (−4 ± 4√3)/4 becomes −1 ± √3.
What is the fastest way to check my answer?
The sum and product of the roots. They must equal −b/a and c/a, and they are read straight from the coefficients rather than from your working.