Guides / Quadratics

Quadratic formula sign mistakes: where they happen and how to catch them

Most wrong answers from the quadratic formula are not arithmetic errors. They are one of six sign errors, each with a specific line it lives on, and each with a quick check.

Step zero: standard form, then name a, b, c with signs

The formula only works on ax² + bx + c = 0. Everything on one side, zero on the other, terms in that order. Then b is the whole coefficient of x, including its sign, and so is c. In x² − 5x + 6 = 0, b = −5, not 5. Writing "a = 1, b = −5, c = 6" as its own line before the formula is the single habit that prevents most of what follows.

Example 1: a negative b

Solve x² − 5x + 6 = 0.

1. a = 1, b = −5, c = 6Name them, with signs.
2. x = −(−5) ± √((−5)² − 4(1)(6))2(1)Substitute with parentheses around every negative. The parentheses are what make the next line come out right.
3. x = 5 ± √(25 − 24)2−(−5) = 5. (−5)² = 25, positive. 4 · 1 · 6 = 24.
4. x = 5 ± 12√1 = 1.
5. x = 3 or x = 2(5 + 1)/2 and (5 − 1)/2.

Check by factoring: (x − 2)(x − 3) = 0. Same roots.

Example 2: a negative c

Solve 2x² + 3x − 5 = 0.

1. a = 2, b = 3, c = −5Name them, with signs.
2. x = −3 ± √(3² − 4(2)(−5))2(2)Substitute.
3. x = −3 ± √(9 + 40)4−4 · 2 · (−5) = +40. Negative c makes the discriminant bigger, never smaller.
4. x = −3 ± 74√49 = 7.
5. x = 1 or x = −52(−3 + 7)/4 = 4/4 and (−3 − 7)/4 = −10/4.

Check x = 1 in the original: 2 + 3 − 5 = 0. True.

Example 3: not in standard form yet

Solve 3x² = 2x + 8.

1. 3x² − 2x − 8 = 0Subtract 2x and 8 from both sides. Both terms change sign as they cross.
2. a = 3, b = −2, c = −8Name them, with signs.
3. x = 2 ± √(4 + 96)6−(−2) = 2. (−2)² = 4. −4(3)(−8) = +96.
4. x = 2 ± 106√100 = 10.
5. x = 2 or x = −4312/6 and −8/6, reduced.

Example 4: a negative discriminant

Solve x² + 2x + 5 = 0.

1. a = 1, b = 2, c = 5Name them, with signs.
2. b² − 4ac = 4 − 20 = −16Compute the discriminant on its own line first.
3. No real solutionsYou can't take the square root of a negative in the real numbers. If your class uses complex numbers: x = (−2 ± 4i)/2 = −1 ± 2i.

A negative discriminant is a legitimate answer, but it is also the most common symptom of a sign error two lines earlier. When you get one, go back and check b² and −4ac before writing "no solution." In Example 1, squaring −5 as −25 would give a discriminant of −49 and a false "no solution" for an equation that factors.

Common mistakes, and the exact line they happen on

1. Reading b without its sign

From x² − 5x + 6 = 0:

1. a = 1, b = 5, c = 6This is the mistake. The x term is −5x, so b = −5. With b = 5 the formula gives (−5 ± 1)/2 = −2 and −3, and the check fails: (−2)² − 5(−2) + 6 = 20, not 0.

2. Squaring a negative and getting a negative

1. b = −5Correct so far.
2. b² = −25This is the mistake. (−5)² = (−5)(−5) = 25. b² is never negative, so a negative number here is always an error.

3. Keeping −b negative when b is negative

1. x = −5 ± √(25 − 24)2This is the mistake. −b means −(−5) = 5. Writing −5 here gives −2 and −3 instead of 3 and 2: right numbers, wrong signs, and the check catches it.

4. Subtracting 4ac when c is negative

From 2x² + 3x − 5 = 0:

1. 9 − 40 = −31This is the mistake. −4ac = −4(2)(−5) = +40, so the discriminant is 9 + 40 = 49. A negative c always adds to the discriminant. Getting a negative discriminant when c is negative is the signal to recheck this line.

5. Dividing only the square root by 2a

1. x = −3 ± 74This is the mistake. The fraction bar goes under the entire numerator: (−3 ± 7)/4. If you write the formula on one line as −b ± √(b² − 4ac) / 2a, this error is almost guaranteed. Draw the long bar.

6. Solving before moving everything to one side

For 3x² = 2x + 8, reading off a = 3, b = 2, c = 8 uses the signs of terms that are on the wrong side. Rearranging to 3x² − 2x − 8 = 0 first gives b = −2, c = −8. Both signs were wrong.

Two checks that take ten seconds

• Plug one root back into the original equation. Not the rearranged one, the original. Any of the six mistakes above makes this fail.
• Sum and product. For ax² + bx + c = 0, the two roots add to −b/a and multiply to c/a. In Example 2, the roots 1 and −5/2 add to −3/2 = −b/a and multiply to −5/2 = c/a. A sign error in b flips the sum; a sign error in c flips the product. This check tells you which letter went wrong.

Practice

1. Solve x² − 7x + 10 = 0.

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   a = 1, b = −7, c = 10. Discriminant 49 − 40 = 9. x = (7 ± 3)/2 = 5 or 2.

2. Solve x² + 4x − 12 = 0.

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   b = 4, c = −12. Discriminant 16 + 48 = 64. x = (−4 ± 8)/2 = 2 or −6.

3. Solve 2x² − 3x − 2 = 0.

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   a = 2, b = −3, c = −2. Discriminant 9 + 16 = 25. x = (3 ± 5)/4 = 2 or −1/2.

4. Solve x² = 6x − 9.

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   Rearrange: x² − 6x + 9 = 0, so b = −6, c = 9. Discriminant 36 − 36 = 0. One repeated root, x = 6/2 = 3.

5. Solve x² − 2x + 5 = 0.

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   Discriminant 4 − 20 = −16. No real solutions (complex: 1 ± 2i). Both signs were right; this one really has none.

6. Solve 3x² + 5x − 2 = 0.

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   a = 3, b = 5, c = −2. Discriminant 25 + 24 = 49. x = (−5 ± 7)/6 = 1/3 or −2. Check the product: (1/3)(−2) = −2/3 = c/a.